Opciones de FX y riesgo de sonrisa
El mercado de opciones FX representa uno de los mercados más líquidos y fuertemente competitivos del mundo, y cuenta con muchas sutilezas técnicas que pueden perjudicar seriamente al comerciante desinformado e ignorante.
Este libro es una guía única para ejecutar un libro de opciones de FX desde la perspectiva del creador de mercado. Logrando un equilibrio entre el rigor matemático y la práctica del mercado y escrito por el experimentado médico Antonio Castagna, el libro muestra a los lectores cómo construir correctamente una superficie de volatilidad completa de los precios de mercado de las estructuras principales.
A partir de las convenciones básicas relacionadas con las principales transacciones de FX y las estructuras básicas negociadas de opciones de FX, el libro introduce gradualmente las principales herramientas para hacer frente al riesgo de volatilidad de FX. A continuación, pasa a revisar los principales conceptos de la teoría de precios de opciones y su aplicación dentro de una economía de Black-Scholes y un entorno de volatilidad estocástica. El libro también presenta los modelos que se pueden implementar para fijar precios y administrar opciones de FX antes de examinar los efectos de la volatilidad sobre los beneficios y pérdidas derivados de la actividad de cobertura.
Cómo se utiliza el modelo Black-Scholes en la actividad comercial profesional
Los modelos de volatilidad estocástica más adecuados
Fuentes de ganancias y pérdidas de la actividad de cobertura de Delta y volatilidad
Conceptos fundamentales de cobertura de sonrisa
Principales enfoques de mercado y variaciones del método Vanna-Volga
Griegos relacionados con la volatilidad en el modelo Black-Scholes
Precios de las opciones simples de vainilla, opciones digitales, opciones de barrera y las opciones exóticas menos conocidas
Herramientas para el monitoreo de los principales riesgos de las opciones FX & # 8217; libro
El libro está acompañado por un CD Rom con modelos en VBA, que demuestra muchos de los enfoques descritos en el libro.
Notación y acrónimos.
1 El mercado de FX.
1.1 Tasas de cambio y contratos spot.
1.2 Contratos de swaps de FX y FX.
1.3 Contratos de opciones de FX.
1.4 Estructuras de opciones FX principales negociadas.
2 Modelos de precios para opciones de FX.
2.1 Principios de la teoría de precios de opciones.
2.2 El modelo negro de los scholes.
2.3 El modelo de Heston.
2.4 El modelo SABR.
2.5 El enfoque de la mezcla.
2.6 Algunas consideraciones sobre la elección del modelo.
3 Hedging dinámico y negociación de la volatilidad.
3.1 Consideraciones preliminares.
3.2 Un marco general.
3.3 Cobertura con volatilidad implícita constante.
3.4 Cobertura con una actualización de volatilidad implícita.
3.5 Hedging Vega.
3.6 Hedging Delta, Vega, Vanna y Volga.
3.7 La sonrisa de la volatilidad y su fenomenología.
3.8 Exposiciones locales a la sonrisa de volatilidad.
3.9 Cobertura del escenario y su relación con la cobertura de Volanna de Vanna.
4 La superficie de volatilidad.
4.1 Definiciones generales.
4.2 Criterios para una representación eficiente y conveniente de la superficie de volatilidad.
4.3 Métodos comúnmente adoptados para construir una superficie de volatilidad.
4.4 Interpolación de sonrisas entre huelgas: el enfoque Vanna & Volga.
4.5 Algunas características del enfoque Vanna & Volga.
4.6 Una caracterización alternativa del enfoque Vanna & Volga.
4.7 Interpolación de sonrisas entre expiries: estructura implícita de términos de volatilidad.
4.8 Superficies de volatilidad admisibles.
4.9 Teniendo en cuenta la mariposa del mercado.
4.10 Construir la matriz de volatilidad en la práctica.
5 opciones de la vainilla llana.
5.1 Precios de opciones simples de vainilla.
5.2 Herramientas para la elaboración del mercado.
5.3 Diferencias bid / ask para opciones simples de vainilla.
5.4 Tiempos de corte y diferenciales.
5.5 Opciones digitales.
5.6 Opciones americanas de vainilla sencilla.
6 Opciones de Barrera.
6.1 Una taxonomía de las opciones de barrera.
6.2 Algunas relaciones de precios de opciones de barrera.
6.3 Precios para las opciones de barrera en una economía BS.
6.4 Fórmulas de precios para opciones de barrera.
6.5 Opciones de un toque (rebate) y sin tocar.
6.6 Opciones de doble barrera.
6.7 Opciones de doble toque y doble toque.
6.8 Probabilidad de golpear una barrera.
6.9 Cálculo griego.
6.10 Opciones de barrera de precios en otros ajustes de modelo.
6.11 Barreras de precios con entrega no estándar.
6.12 Aproximación al mercado de las opciones de barreras de precios.
6.13 Diferencias de pujas / ofertas.
6.14 Frecuencia de monitoreo.
7 Otras opciones exóticas.
7.2 Opciones de barrera a la expiración.
7.3 Opciones de barrera de ventana.
7.4 Las primeras opciones de barrido knock-in y knock-in.
7.5 Opciones Auto-quanto.
7.6 Opciones de inicio directo.
7.7 Swaps de desviación.
7.8 Compuesto, asiático y opciones de lookback.
8 Herramientas y análisis de gestión de riesgos.
8.2 Implementación del modelo LMUV.
8.3 Herramientas de seguimiento de riesgos.
8.4 Análisis de riesgo de opciones de vainilla simple.
8.5 Análisis de riesgos de las opciones digitales.
9 Correlación y Opciones de FX.
9.1 Consideraciones preliminares.
9.2 Correlación en el ajuste BS.
9.3 Contratos dependiendo de varios tipos de cambio FX.
9.4 Lidiar con la correlación y volatilidad sonrisa.
9.5 Vinculación de las sonrisas de la volatilidad.
Admin Trial 32 comentarios
El mercado de opciones FX representa uno de los mercados más líquidos y fuertemente competitivos del mundo, y presenta muchas sutilezas técnicas que pueden seriamente. Forschungsgemeinschaft a través del SFB 649 "Riesgo Económico". La sonrisa de las opciones de vainilla se puede reproducir calibrando adecuadamente tres de cada cinco modelos. PDFx. GBM. Heston. -2. -1. 0. 1. 2. 10-8. 10-6. 10-4. 10-2. 100 x. PDFx. Las opciones de acciones negociadas en los mercados estadounidenses no mostraban antes una sonrisa de volatilidad. Sin embargo, las volatilidades implícitas de las opciones sobre contratos de divisas tienden. Las reversiones del riesgo se cotizan generalmente como la inversión del riesgo del delta del X% y es esencialmente. Volatilidad Modelado de Sonrisas con Ecuaciones Diferenciales Estocásticas de Mezcla.
Fx opción y riesgo sonrisa pdf
VANILLA FOREX OPCIONES GARMAN-KOHLHAGEN Y RIESGO. La sonrisa usando las cotizaciones estándar del mercado en-el-dinero, la inversión del riesgo y el estrangulamiento es. El mercado de opciones FX representa uno de los mercados más líquidos y fuertemente competitivos del mundo, y presenta muchas sutilezas técnicas que pueden seriamente. Cotizaciones para cada madurez de mercado el 0 Δ straddle, la inversión de riesgo y el vega-. Cedimiento, también llamado Vanna-Volga VV, para construir la sonrisa En el mercado de opciones de FX, la matriz de volatilidad se construye de acuerdo con la regla de Delta pegajosa.
Red de comercio de la tierra twitter
Y las sonrisas correctamente es fundamental para los mostradores de renta fija y divisas, ya que éstos. Un riesgo de + $ 1MM, y es larga baja huelga opciones con un A de - $ 1MM. La sonrisa. Aplicamos el modelo SABR a las opciones de tipos de interés de USD, y encontramos buenas. De renta fija y de divisas, ya que estos escritorios tienen generalmente.
Grandes acciones
Mar 29, 2004. Los precios de opción de FX se pueden utilizar para entender FX neutral de riesgo. Dar, y hay una sonrisa en la opción de FX implieds, es decir, la medida de convexidad. La C de su huelga K se obtiene integrando dos veces el π de π de π de. 5 de enero de 2006. En los mercados actuales, las opciones con diferentes huelgas o vencimientos suelen tener un precio con diferentes volatilidades implícitas. La inversión del riesgo y la mariposa ponderada por vegas, presentándonos así el problema. Número de páginas en archivo PDF 15. Palabras clave Opción FX, sonrisa, consistencia de precios, volatilidad estocástica.
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Opciones de FX y riesgo de sonrisa (eBook, PDF)
Opciones de FX y riesgo de sonrisa (eBook, PDF)
El mercado de opciones de FX representa uno de los mercados más líquidos y altamente competitivos del mundo, y presenta muchas sutilezas técnicas que pueden perjudicar seriamente al desactualizado comerciante de atún. Este libro es una guía única para ejecutar un libro de opciones de FX desde la perspectiva del creador de mercado. Al encontrar un equilibrio entre el rigor matemático y la práctica del mercado y escrito por el experimentado experto Antonio Castagna, el libro muestra a los lectores cómo construir correctamente una superficie de volatilidad completa de los precios de mercado de las estructuras principales. Comenzando con los convenios básicos relacionados con el principal FX & hellip; mehr
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El mercado de opciones de FX representa uno de los mercados más líquidos y altamente competitivos del mundo, y presenta muchas sutilezas técnicas que pueden perjudicar seriamente al desactualizado comerciante de atún.
Este libro es una guía única para ejecutar un libro de opciones de FX desde la perspectiva del creador de mercado. Al encontrar un equilibrio entre el rigor matemático y la práctica del mercado y escrito por el experimentado experto Antonio Castagna, el libro muestra a los lectores cómo construir correctamente una superficie de volatilidad completa de los precios de mercado de las estructuras principales.
Comenzando con las convenciones básicas relacionadas con las principales ofertas de divisas y las estructuras básicas negociadas de opciones de cambio, el libro introduce gradualmente las principales herramientas para hacer frente al riesgo de volatilidad de divisas. Itthen pasa a revisar los conceptos principales de la teoría de la fijación de precios de opciones y su aplicación dentro de una economía de Black-Scholes y un entorno de volatilidad astocástica. El libro también introduce modelos que se pueden implementar para fijar precios y administrar las opciones de divisas antes de analizar los efectos de la volatilidad sobre los beneficios y la pérdida de la actividad de cobertura.
* Cómo se utiliza el modelo de Black-Scholes en la actividad de negociación profesional * los modelos de volatilidad estocástica más adecuados * fuentes de ganancias y pérdidas de la actividad de Delta y volatilidad * conceptos fundamentales de cobertura de sonrisa * enfoques de mercado y variaciones importantes de volatilidad Vanna-Volgamethod * * Los precios de las opciones simples de vainilla, opciones digitales, barreras y las opciones exóticas menos conocidas * herramientas para el monitoreo de los principales riesgos de un libro de opciones FX
El libro está acompañado por un CD Rom con modelos en VBA, que demuestra muchos de los enfoques descritos en el libro.
Antonio Castagna es actualmente socio y cofundador de la consultora Iason ltd, prestando apoyo a las instituciones financieras para el diseño de modelos de precios de derivados complejos y para medir una amplia gama de riesgos, incluyendo el crédito y la liquidez. Antonio se graduó en Finanzas de la Universidad LUISS de Roma en 1995 con una tesis sobre opciones americanas y los procedimientos numéricos para su valoración. Comenzó su carrera en banca de inversión en IMI Bank, Luxemborug, como analista financiero en el Departamento de Control de Riesgos antes de trasladarse a Banca IMI, Milán, primero como un creador de mercado de capital / piso e intercambios, antes de configurar la mesa de opciones FX y Ejecutando el libro de la vainilla llana y las opciones exóticas en las monedas principales, mientras que también siendo responsable de la negociación entera de la volatilidad de FX. Antonio ha escrito una serie de artículos sobre derivados de crédito, gestión de riesgos de opciones exóticas y sonrisas de volatilidad. A menudo es invitado a cursos académicos y de posgrado.
Notación y acrónimos.
1 El mercado de FX.
1.1 Tasas de cambio y contratos spot.
1.2 Contratos de swaps de FX y FX.
1.3 Contratos de opciones de FX.
1.4 Estructuras de opciones FX principales negociadas.
2 Modelos de precios para opciones de FX.
2.1 Principios de la teoría de precios de opciones.
2.2 El modelo negro-scholes.
2.3 El modelo de Heston.
2.4 El modelo SABR.
2.5 El enfoque de la mezcla.
2.6 Algunas consideraciones sobre la elección del modelo.
3 Hedging dinámico y negociación de la volatilidad.
3.1 Consideraciones preliminares.
3.2 Un marco general.
3.3 Cobertura con volatilidad implícita constante.
3.4 Cobertura con una actualización de volatilidad implícita.
3.5 Hedging Vega.
3.6 Hedging Delta, Vega, Vanna y Volga.
3.7 La sonrisa de la volatilidad y su fenomenología.
3.8 Exposiciones locales a la sonrisa de volatilidad.
3.9 Cobertura del escenario y su relación con la cobertura de Vanna-Volga.
4 La superficie de volatilidad.
4.1 Definiciones generales.
4.2 Criterios para una representación eficiente y conveniente de la superficie de volatilidad.
4.3 Métodos comúnmente adoptados para construir una superficie de volatilidad.
4.4 Interpolación de sonrisas entre huelgas: el enfoque Vanna-Volga.
4.5 Algunas características del enfoque Vanna-Volga.
4.6 Una caracterización alternativa del enfoque Vanna-Volga.
4.7 Interpolación de sonrisas entre expiries: estructura implícita de términos de volatilidad.
4.8 Superficies de volatilidad admisibles.
4.9 Teniendo en cuenta la mariposa del mercado.
4.10 Construir la matriz de volatilidad en la práctica.
5 opciones de la vainilla llana.
5.1 Precios de opciones simples de vainilla.
5.2 Herramientas para la elaboración del mercado.
5.3 Diferencias bid / ask para opciones simples de vainilla.
5.4 Tiempos de corte y diferenciales.
5.5 Opciones digitales.
5.6 Opciones americanas de vainilla sencilla.
6 Opciones de Barrera.
6.1 Una taxonomía de las opciones de barrera.
6.2 Algunas relaciones de precios de opciones de barrera.
6.3 Precios para las opciones de barrera en una economía BS.
6.4 Fórmulas de precios para opciones de barrera.
6.5 Opciones de un toque (rebate) y sin tocar.
6.6 Opciones de doble barrera.
6.7 Opciones de doble toque y doble toque.
6.8 Probabilidad de golpear una barrera.
6.9 Cálculo griego.
6.10 Opciones de barrera de precios en otros ajustes de modelo.
6.11 Barreras de precios con entrega no estándar.
6.12 Aproximación al mercado de las opciones de barreras de precios.
6.13 Diferencias de pujas / ofertas.
6.14 Frecuencia de monitoreo.
7 Otras opciones exóticas.
7.2 Opciones de barrera a la expiración.
7.3 Opciones de barrera de ventana.
7.4 Opciones de barrera en primer lugar y en caso de knock-in-knock-out.
7.5 Opciones Auto-quanto.
7.6 Opciones de inicio directo.
7.7 Swaps de desviación.
7.8 Compuesto, asiático y opciones de lookback.
8 Herramientas y análisis de gestión de riesgos.
8.2 Implementación del modelo LMUV.
8.3 Herramientas de control de riesgos.
8.4 Análisis de riesgo de opciones de vainilla simple.
8.5 Análisis de riesgos de las opciones digitales.
9 Correlación y Opciones de FX.
9.1 Consideraciones preliminares.
9.2 Correlación en el ajuste BS.
9.3 Contratos dependiendo de varios tipos de cambio FX.
9.4 Lidiar con la correlación y volatilidad sonrisa.
9.5 Vinculación de las sonrisas de la volatilidad.
Precios de Opción de Cambio de Divisas: Guía de un Practicante
Este libro cubre las opciones de divisas desde el punto de vista del profesional de las finanzas. Contiene todo lo que un cuatrero o comerciante que trabaja en un banco o un fondo de cobertura necesitaría saber acerca de las matemáticas de divisas, no sólo las matemáticas teóricas cubiertas en otros libros, sino también una cobertura completa de la implementación, la fijación de precios y la calibración.
Con contenidos desarrollados con aportaciones de comerciantes y con ejemplos que utilizan datos reales, este libro presenta muchos de los productos más solicitados de las mesas de operaciones de opciones de FX, junto con los modelos que capturan las características de riesgo necesarias para tasar estos productos con precisión. Crucially, este libro describe los métodos numéricos requeridos para la calibración de estos modelos & ndash; Un área que a menudo se descuida en la literatura, que es sin embargo de importancia primordial en la práctica. El tratamiento completo se da en un texto unificado a las siguientes características:
Convenciones de mercado correctas para la superficie de volatilidad FX
Ajuste para liquidación y entrega retrasada de opciones
Precios de vainillas y opciones de barrera bajo la sonrisa volatilidad
Curvatura de barrera para limitar el riesgo de discontinuidad de barrera cerca de la fecha de caducidad
Ecuaciones diferenciales parciales de resistencia industrial en una y varias variables espaciales usando diferencias finitas en grillas no uniformes
Métodos de transformación de Fourier para fijar el precio de las opciones europeas mediante funciones características
Modelos estocásticos y de volatilidad local, y un modelo mixto estocástico / volatilidad local
Modelo de FX de tres factores de larga data
Técnicas de calibración numérica para todos los modelos de este trabajo
El enfoque de variable de estado aumentada para fijar el precio de las opciones fuertemente dependientes de la trayectoria utilizando ecuaciones diferenciales parciales o simulación de Monte Carlo
Conectando la teoría matemáticamente rigurosa con la práctica, ésta es la guía esencial de las opciones de divisas en el contexto del mercado financiero real.
Lista de tablas xv
Lista de figuras xvii
1 Introducción 1
1.1 Una suave introducción a los mercados de divisas 1
1.2 Estilos de cotización 2
1.3 Consideraciones sobre el riesgo 5
1.4 Reglas de Liquidación Spot 5
1.5 Normas de expiración y entrega 8
1.5.1 Normas de caducidad y de entrega & ndash; Días o semanas 8
1.5.2 Requisitos de caducidad y de entrega & ndash; Meses o años 9
1.6 Tiempos de corte 10
2 Preliminares Matemáticas 13
2.1 El modelo Black & ndash; Scholes 13
2.1.1 Supuestos del modelo Black de Scholes 13
2.2 Neutralidad del riesgo 13
2.3 Derivación de la ecuación de Black & # 8217; Scholes 14
2.4 Integración del SDE para ST 17
2.5 Black & ndash; Scholes PDEs expresadas en Logspot 18
2.6 Feynman & ndash; Kac y Expectativa de riesgo-neutral 18
2.7 Neutralidad del riesgo y presunción de deriva 20
2.8 Valoración de las opciones europeas 23
2.9 La ley del precio único 27
2.10 El modelo de estructura de término Black & ndash; Scholes 28
2.11 Breeden - Análisis de Litzenberger 30
2.12 Digitales europeos 31
2.13 Ajustes de liquidación 32
2.14 Ajustes de entrega retrasados 33
2.15 Precios utilizando Métodos de Fourier 35
2.15.1 Precio de opción europeo que implica una integral numérica 37
2.16 Leptokurtosis & ndash; Más que Fat Tails 38
3 Deltas y convenciones de mercado 41
3.1 Conversiones de estilo de cita 41
3.2 La Ley de muchos Deltas 43
3.3 Convenciones FX Delta 47
3.4 Volatilidad del mercado Superficies 49
3.5 En el dinero 50
3.6 Estrangulamiento del mercado 53
3.6.1 Ejemplo & ndash; EURUSD 1Y 55
3.7 Sonrisa estrangulamiento y reversión del riesgo 55
3.8 Visualización de los Strangles 57
3.9 Interpolación de sonrisa & ndash; Polinomio en Delta 59
3.10 Interpolación de sonrisa & ndash; SABR 60
3.11 Observaciones finales 62
4 Volatilidad Construcción Superficial 63
4.1 Volatilidad Backbone & ndash; Interpolación directa plana 65
4.2 Volatilidad Interpolación temporal de la superficie 67
4.3 Volatilidad Interpolación Temporal Superficial & ndash; Vacaciones y fines de semana 70
4.4 Volatilidad Interpolación Temporal Superficial & ndash; Efectos Intradía 73
5 Volatilidad local y volatilidad implícita 77
5.1 Introducción 77
5.2 La ecuación de Fokker Planck 78
5.3 Construcción de la volatilidad local de Dupire 83
5.4 Volatilidad implícita y relación con la volatilidad local 86
5.5 La volatilidad local como expectativa condicional 87
5.6 Volatilidad local de los mercados de divisas 88
5.7 Difusión y PDE para la volatilidad local 89
5.8 El modelo CEV 90
5.8.1 Expansión asintótica 91
6 Volatilidad estocástica 95
6.1 Introducción 95
6.2 Volatilidad incierta 95
6.3 Modelos de volatilidad estocástica 96
6.4 Volatilidad estocástica no correlacionada 107
6.5 Volatilidad estocástica correlacionada con el punto 108
6.6 El Plan Fokker Planck PDE Approach 111
6.7 El enfoque Feynman Kac PDE 113
6.8 Modelos de Volatilidad Estocástica Local (LSV) 117
7 Métodos Numéricos de Precios y Calibración 129
7.1 Búsqueda de Raíz Unidimensional & ndash; Cálculo implícito de volatilidad 129
7.2 Minimización de mínimos cuadrados no lineales 130
7.3 Simulación de Monte Carlo 131
7.4 Convección PDEs de difusión en finanzas 147
7.5 Métodos numéricos para PDE 153
7.6 Esquema de Diferencia Finita Explicada 155
7.7 Diferencia finita explícita en mallas no uniformes 163
7.8 Esquema de Diferencia Finita Implícita 165
7.9 El sistema Crank & Nicolson 167
7.10 Esquemas numéricos para PDEs multidimensionales 168
7.11 Esquemas prácticos de generación de cuadrículas no uniformes 173
7.12 Otras lecturas 176
8 Exotics de primera generación & ndash; Opciones binarias y de barrera 177
8.1 El Principio de Reflexión 179
8.2 Barreras y binarios europeos 180
8.3 Binarios y barreras monitoreados continuamente 183
8.4 Productos de doble barrera 194
8.5 Sensibilidad a la volatilidad local y estocástica 195
8.6 Doblado de la barrera 197
8.7 Supervisión del valor 202
9 Exóticas de Segunda Generación 205
9.1 Opciones del Selector 206
9.2 Opciones de acumulación de rango 206
9.3 Opciones de inicio directo 207
9.4 Opciones de retroceso 209
9.5 Opciones asiáticas 212
9.6 Notas de reembolso objetivo 214
9.7 Swaps de volatilidad y variación 214
10 Opciones Multicurrency 225
10.1 Correlaciones, triangulación y ausencia de arbitraje 226
10.2 Opciones de intercambio 229
10.3 Quantos 229
10.4 Lo mejor de lo peor 233
10.5 Opciones de la cesta 239
10.6 Métodos Numéricos 241
10.7 Nota sobre los Multicurrency Greeks 242
10.8 Determinación de factores no transaccionables 243
10.9 Otras lecturas 244
11 Longdated FX 245
11.1 Swaps de divisas 245
11.2 Riesgo básico 247
11.3 Medida futura 249
11.4 LIBOR en atrasos 250
11.5 Productos típicos de larga duración FX 253
11.6 El modelo de tres factores 255
11.7 Calibración de la tasa de interés del modelo de tres factores 257
11.8 Calibración Spot FX del Modelo de Tres Factores 259
11.9 Conclusión 264
Lectura adicional 271
Dr. Iain J. Clark. (Londres, Reino Unido), es Jefe de Análisis Cuantitativo de Divisas en Dresdner Kleinwort en Londres, donde creó y dirige el equipo responsable de desarrollar las bibliotecas de precios para la oficina. Anteriormente, fue Director del Grupo de Investigación Cuantitativa de Lehman Brothers, Analista cuantitativo de renta fija de BNP Paribas y también trabajó en la investigación de derivados de materias primas FX en JP Morgan. Tiene una Maestría en Matemáticas de la Universidad de Edimburgo y un Doctorado en Matemáticas Aplicadas de la Universidad de Queensland, Australia. El Dr. Clark es orador habitual en eventos clave de finanzas, y ha presentado en el Imperial College de Londres, la Conferencia Anual de la Sociedad Bachelier, el Imperial College de Londres, la Conferencia anual de Estrategias de negocios mundiales, eventos de riesgo, eventos Marcus Evans y muchos más.
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Opciones de FX y riesgo de sonrisa
Opciones de FX y Riesgo de Sonrisas por Antonio Castagna 2010 | ISBN: 0470754192 | Español | 330 páginas | PDF | 3 MB El mercado de opciones de FX representa uno de los mercados más líquidos y fuertemente competitivos del mundo, y cuenta con muchas sutilezas técnicas que pueden perjudicar gravemente al comerciante desinformado y no consciente. Este libro es una guía única para ejecutar un libro de opciones de FX desde la perspectiva del creador de mercado. Logrando un equilibrio entre el rigor matemático y la práctica del mercado y escrito por el experimentado médico Antonio Castagna, el libro muestra a los lectores cómo construir correctamente una superficie de volatilidad de los precios de mercado de las estructuras principales.
A partir de las convenciones básicas relacionadas con las principales transacciones de FX y las estructuras básicas negociadas de opciones de FX, el libro introduce gradualmente las principales herramientas para hacer frente al riesgo de volatilidad de FX. A continuación, pasa a revisar los principales conceptos de la teoría de precios de opciones y su aplicación dentro de una economía de Black-Scholes y un entorno de volatilidad estocástica. El libro también presenta los modelos que se pueden implementar para fijar precios y administrar opciones de FX antes de examinar los efectos de la volatilidad sobre los beneficios y pérdidas derivados de la actividad de cobertura.
Cómo el modelo de Black-Scholes se utiliza en la actividad comercial profesional los modelos de volatilidad estocástica más adecuados fuentes de ganancias y pérdidas de la actividad de cobertura de Delta y volatilidad conceptos fundamentales de cobertura de sonrisa enfoques de mercado y variaciones importantes del método Vanna Volga griegos relacionados con la volatilidad En el modelo de precios Black-Scholes de las opciones simples de vainilla, las opciones digitales, las opciones de barrera y las herramientas de opciones exóticas menos conocidas para monitorear los principales riesgos de un libro de opciones FX El libro está acompañado por un CD ROM con modelos en VBA, Muchos de los enfoques descritos en el libro.
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Por otra parte, en los llamados modelos de volatilidad estocástica, la volatilidad σ t es modelada como una semi-martingala browniana continua, entre los modelos de volatilidad estocástica se encuentran el modelo Hull y White [32], el modelo de Heston [31] , Y el modelo SABR [29], mientras que la dinámica de la volatilidad estocástica son más realistas que la dinámica de la volatilidad local, los precios de las opciones generadas no son coherentes con los precios de las opciones europeas observadas.
Artículo: La volatilidad es áspera
RESUMEN: Estimando la volatilidad a partir de datos recientes de alta frecuencia, se revisa la cuestión de la suavidad del proceso de volatilidad. Nuestro resultado principal es que la log-volatilidad se comporta esencialmente como un movimiento browniano fraccionario con el exponente de Hurst H de orden 0.1, a cualquier escala de tiempo razonable. Esto nos lleva a adoptar el modelo de volatilidad estocástica fraccional (FSV) de Comte y Renault. Llamamos a nuestro modelo Rough FSV (RFSV) para subrayar que, en contraste con FSV, H & lt; 1/2. Demostramos que nuestro modelo RFSV es notablemente consistente con los datos de series temporales financieras; Una aplicación es que nos permite obtener mejores pronósticos de volatilidad realizada. Además, encontramos que aunque la volatilidad no es memoria larga en el modelo RFSV, los procedimientos estadísticos clásicos con el objetivo de detectar la persistencia de la volatilidad tienden a concluir la presencia de memoria larga en los datos generados a partir de ella. Esto arroja luz sobre por qué el recuerdo de la volatilidad ha sido ampliamente aceptado como un hecho estilizado. Por último, ofrecemos una base cuantitativa basada en la microestructura del mercado para nuestros hallazgos, relacionando la aspereza de la volatilidad con el comercio de alta frecuencia y la división de pedidos.
Texto completo & middot; Artículo & middot; Oct 2017
Disponible en: Patrick S Hagan
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MANAGING SMILE RISK PATRICK S. HAGAN *, PROFUNDO KUMAR †, ANDREW S. LESNIEWSKI ‡, Y DIANA E. WOODWARD§ Resumen. Las sonrisas de mercado y las distorsiones se manejan generalmente usando modelos locales de la volatilidad a la Dupire. Descubrimos que la dinámica de la sonrisa del mercado pronosticada por los modelos de vol locales es opuesta al comportamiento observado del mercado: cuando el precio del subyacente disminuye, los modelos de vol local predicen que la sonrisa cambia a precios más altos; Cuando el precio aumenta, estos modelos predicen que la sonrisa cambia a precios más bajos. Debido a esta contradicción entre el modelo y el mercado, los setos delta y vega derivados del modelo pueden ser inestables y pueden tener un comportamiento peor que los setos de Black-Scholes ingenuos. Para eliminar este problema, derivamos el modelo SABR, un modelo de volatilidad estocástica en el que el valor forward satisface? =?. 1 \ alpha. =?. 2 y el delantero? Y volatilidad. Están correlacionados. 1 ^ {2} =. Utilizamos técnicas de perturbación singulares para obtener los precios de las opciones europeas bajo el modelo SABR, ya partir de estos precios obtenemos fórmulas algebraicas explícitas de forma cerrada para la volatilidad implícita como funciones del precio a plazo de hoy. =? (0) y la huelga. Estas fórmulas producen inmediatamente el precio de mercado, los riesgos de mercado, incluyendo los riesgos de vanna y volga, y muestran que el modelo SABR captura la dinámica correcta de la sonrisa. Aplicamos el modelo SABR a las opciones de tipos de interés de USD, y encontramos un buen acuerdo entre las sonrisas teóricas y observadas. Palabras claves. Sonrisas, sesgo, cobertura dinámica, stochastic vols, volga, vanna 1. Introducción. Las opciones europeas suelen ser tasadas y cubiertas usando el modelo de Black o, de forma equivalente, el modelo de Black-Scholes. En el modelo de Black hay una relación uno a uno entre el precio de una opción europea y el parámetro de volatilidad. Por lo tanto, los precios de las opciones suelen ser citados al declarar la volatilidad implícita. El valor único de la volatilidad que produce el precio en dólares de la opción cuando se usa en el modelo de Black. En teoría, la volatilidad. En el modelo de Black es una constante. En la práctica, ¿opciones con diferentes huelgas? Requieren diferentes volatilidades. Para igualar sus precios de mercado. Véase la figura 1. El manejo correcto de estas distorsiones y sonrisas del mercado es crítico para los mostradores de renta fija y divisas, ya que estos escritorios suelen tener grandes exposiciones en una amplia gama de huelgas. Sin embargo, la contradicción inherente de utilizar diferentes volatilidades para diferentes opciones dificulta el manejo exitoso de estos riesgos utilizando el modelo de Black. El desarrollo de modelos de volatilidad local por Dupire [2], [3] y Derman-Kani [4], [5] fue un avance importante en el manejo de sonrisas y sesgos. Los modelos de volatilidad local son autoconstantes, libres de arbitraje y pueden calibrarse para adaptarse con precisión a las sonrisas y distorsiones observadas en el mercado. Actualmente estos modelos son la manera más popular de manejar el riesgo de sonrisa y sesgo. Sin embargo, como se verá en la sección 2, el comportamiento dinámico de las sonrisas y los sesgos predichos por los modelos de vol locales es exactamente opuesto al comportamiento observado en el mercado: cuando el precio del activo subyacente disminuye, los modelos locales de vol predicen que la sonrisa cambia a los precios más altos; Cuando el precio aumenta, estos modelos predicen que la sonrisa cambia a precios más bajos. En realidad, los precios de los activos y las sonrisas de mercado se mueven en la misma dirección. Esta contradicción entre el modelo y el mercado tiende a desestabilizar las coberturas delta y vega derivadas de los modelos de volatilidad local, ya menudo estas coberturas se comportan peor que las coberturas ingenuas de Black-Scholes. Para resolver este problema, derivamos el modelo SABR, un modelo de volatilidad estocástica en el que el precio de los activos y la volatilidad están correlacionados. Se utilizan técnicas singulares de perturbación para obtener los precios de las opciones europeas bajo el modelo SABR, ya partir de estos precios se obtiene una fórmula algebraica de forma cerrada para la volatilidad implícita en función del precio a plazo de hoy. Y la huelga. Esta fórmula cerrada para la volatilidad implícita permite que el precio de mercado y los riesgos de mercado, incluyendo los riesgos de vanna y volga, sean *phagan@bear. com; Bear-Stearns Inc. 383 Madison Avenue, Nueva York, NY 10179 † BNP Paribas; 787 Seventh Avenue; Nueva York NY 10019 ‡ BNP Paribas; 787 Seventh Avenue; Nueva York NY 10019 §Societe Generale; 1221 Avenida de las Américas; Nueva York NY 10020 1
Obtenido inmediatamente de la fórmula de Black. También proporciona buenas, ya veces espectaculares, ajustes a las curvas de volatilidad implícitas observadas en el mercado. Véase la figura 1.1. Más importante aún, la fórmula muestra que el modelo SABR capta la dinámica correcta de la sonrisa, y por lo tanto, produce setos estables. M99 Opción Eurodólar 5 10 15 20 25 30 92,0 93,094.0 95,0 96,097,0 Vol. Vol. (%) Fig. 1.1. Volatilidad implícita para las opciones del Eurodólar de junio de 1999. Se muestran los valores de cierre de día junto con las volatilidades predichas por el modelo SABR. Datos tomados de los servicios de información de Bloomberg el 23 de marzo de 1999. 2. Reprise. Considere una opción de compra europea en un activo A con fecha de ejercicio. fecha de liquidación. Y huelga. Si el titular ejerce la opción en. Entonces en la fecha de liquidación. Recibe el activo subyacente A y paga la huelga. To derive the value of the option, defineˆ?(?) to be the forward price of the asset for a forward contract that matures on the settlement date. and define. =ˆ?(0) to be today’s forward price. Also let ?(?) be the discount factor for date ?; that is, let ?(?) be the value today of $1 to be delivered on date. In Appendix A the fundamental theorem of arbitrage free pricing [6], [8] is used to develop the theoretical framework for European options. There it is shown that the value of the call option is n and the value of the corresponding European put is n ≡. + ?(. )[? − ?]? Here the expectation. is over the forward measure, and “|F0” can be interpretted as “given all information available at. = 0.” See Appendix A. In Appendix A it is also shown that the forward priceˆ?(?) is a Martingale under the forward measure. Therefore, the Martingale representation theorem implies thatˆ?(?) evolves according to (2.1a) . = ?(. ) ?[ˆ?(. ) − ?]+|F0 o ? . = ?(. )? [? −ˆ?(. )]+|F0 o (2.1b) (2.1c)?ˆ? = ?(??∗). ˆ?(0) =. 2
for some coefficient ?(??∗), where. is Brownian motion in this measure. The coefficient ?(??∗) may be deterministic or random, and may depend on any information that can be resolved by time. This is as far as the fundamental theory of arbitrage free pricing goes. In particular, one cannot determine the coefficient ?(??∗) on purely theoretical grounds. Instead one must postulate a mathematical model for ?(??∗)? European swaptions fit within an indentical framework. Consider a European swaption with exercise date. and fixed rate (strike). Let. (?) be the swaption’s forward swap rate as seen at date. and let ?0=ˆ??(0) be the forward swap rate as seen today. In Appendix A we show that the value of a payer swaption is n and the value of a receiver swaption is n ≡. + ?0[. − ?0]? (2.2a) . = ?0? [ˆ??(. ) −. ]+|F0 o ? . = ?0? [. −ˆ??(. )]+|F0 o (2.2b) Here ?0is today’s value of the level (annuity), which is a known quantity, and. is the expectation over the level measure of Jamshidean [10]. In Appendix A it is also shown that the forward swap rateˆ??(?) is a Martingale in this measure, so once again (2.2c)?ˆ??= ?(??∗). ˆ??(0) = ?0? dónde. is Brownian motion. As before, the coefficient ?(??∗) may be deterministic or random, and cannot be determined from fundamental theory. Apart from notation, this is identical to the framework provided by equations 2.1a - 2.1c for European calls and puts. Caplets and floorlets can also be included in this picture, since they are just one period payer and receiver swaptions. For the remainder of the paper, we adopt the notation of 2.1a - 2.1c for general European options. 2.1. Black’s model and implied volatilities. To go any further requires postulating a model for the coefficient ?(??∗). In [11], Black postulated that the coefficient ?(??∗) is. ˆ?(?), where the volatilty. is a constant. The forward priceˆ?(?) is then geometric Brownian motion: (2.3)?ˆ? =. ˆ?(?). ˆ?(0) =. Evaluating the expected values in 2.1a, 2.1b under this model then yields Black’s formula, . = ?(. ) ? . =. + ?(. )[? − ?]? (2.4a) (2.4b) where (2.4c) ?1?2=log. ±1 2?2 . ??√. ? for the price of European calls and puts, as is well-known [11], [12], [13]. All parameters in Black’s formula are easily observed, except for the volatility. An option’s implied volatility is the value of. that needs to be used in Black’s formula so that this formula matches the market price of the option. Since the call (and put) prices in 2.4a - 2.4c are increasing functions of. the volatility ??implied by the market price of an option is unique. Indeed, in many markets it is standard practice to 3
quote prices in terms of the implied volatility. ; the option’s dollar price is then recovered by substituting the agreed upon. into Black’s formula. The derivation of Black’s formula presumes that the volatility. is a constant for each underlying asset A. However, the implied volatility needed to match market prices nearly always varies with both the strike ? and the time-to-exercise. See figure 2.1. Changing the volatility. means that a different model is being used for the underlying asset for each. y. This causes several problems managing large books of options. The first problem is pricing exotics. Suppose one needs to price a call option with strike ?1which has, say, a down-and-out knock-out at ?2. 1. Should we use the implied volatility at the call’s strike ?1, the implied volatility at the barrier ?2, or some combination of the two to price this option? Clearly, this option cannot be priced without a single, self-consistent, model that works for all strikes without “adjustments.” 0.18 0.20 0.22 0.24 0.26 0.28 8090 100110 120 Strike Vol 1m 3m 6m 12m Fig. 2.1. Implied volatility. (?) as a function of the strike. for 1 month, 3 month, 6 month, and 12 month European options on an asset with forward price 100. The second problem is hedging. Since different models are being used for different strikes, it is not clear that the delta and vega risks calculated at one strike are consistent with the same risks calculated at other strikes. For example, suppose that our 1 month option book is long high strike options with a total ∆ risk of +$1. and is long low strike options with a ∆ of −$1. Is our is our option book really ∆-neutral, or do we have residual delta risk that needs to be hedged? Since different models are used at each strike, it is not clear that the risks offset each other. Consolidating vega risk raises similar concerns. Should we assume parallel or proportional shifts in volatility to calculate the total vega risk of our book? More explicitly, suppose that. is 20% at. = 100 and 24% at. = 90, as shown for the 1m options in figure 2.1. Should we calculate vega by bumping. by, say, 0?2% for both options? Or by bumping. by 0?2% for the first option and by 0?24% for the second option? These questions are critical to effective book management, since this requires consolidating the delta and vega risks of all options on a given asset before hedging, so that only the net exposure of the book is hedged. Clearly one cannot answer these questions without a model that works for all strikes. The third problem concerns evolution of the implied volatility curve. (?). Since the implied volatility 4
depends on the strike. it is likely to also depend on the current value. of the forward price. = ??(. ). In this case there would be systematic changes in. as the forward price. of the underlying changes See figure 2.1. Some of the vega risks of Black’s model would actually be due to changes in the price of the underlying asset, and should be hedged more properly (and cheaply) as delta risks. 2.2. Local volatility models. An apparent solution to these problems is provided by the local volatil - ity model of Dupire [2], which is also attributed to Derman [4], [5]. In an insightful work, Dupire essentially argued that Black was too bold in setting the coefficient ?(??∗) to. ˆ. Instead one should only assume that. is Markovian. = ?(??ˆ?). Re-writing ?(??ˆ?) as. (??ˆ?)ˆ? then yields the “local volatility model,” where the forward price of the asset is (2.5a)?ˆ? =. (??ˆ?)ˆ. ˆ?(0) =. in the forward measure. Dupire argued that instead of theorizing about the unknown local volatility function . (??ˆ?), one should obtain. (??ˆ?) directly from the marketplace by “calibrating” the local volatility model to market prices of liquid European options. In calibration, one starts with a given local volatility function. (??ˆ?), and evaluates n ≡. + ?(. )(? − ?) to obtain the theoretical prices of the options; one then varies the local volatility function. (??ˆ?) until these theoretical prices match the actual market prices of the option for each strike. and exercise date . In practice liquid markets usually exist only for options with specific exercise dates ?1 example, for 1m, 2m, 3m, 6m, and 12m from today. Commonly the local vols. (??ˆ?) are taken to be piecewise constant in time: . = ?(. )? [ˆ?(. ) − ?]+|ˆ?(0) =. o (2.5b) (2.5c) . 2. 3. ; for . (??ˆ?) = ?(1) . (??ˆ?) = ?(?) . (??ˆ?) = ?(?) . (ˆ?) for. 1. . (ˆ?) for. −1 ?? . ? ? = 2?3. (2.6) . (ˆ?)for. ? One first calibrates ?(1) reproduce the option prices at ?2 by using the results in [14] and [15]. There we solve to obtain the prices of European options under the local volatility model 2.5a - 2.5c, and from these prices we obtain explicit algebraic formulas for the implied volatility of the local vol models. Once. (??ˆ?) has been obtained by calibration, the local volatility model is a single, self-consistent model which correctly reproduces the market prices of calls (and puts) for all strikes. and exercise dates . without “adjustment.” Prices of exotic options can now be calculated from this model without ambiguity. This model yields consistent delta and vega risks for all options, so these risks can be consolidated across strikes. Finally, perturbing. and re-calculating the option prices enables one to determine how the implied volatilites change with changes in the underlying asset price. Thus, the local volatility model provides a method of pricing and hedging options in the presence of market smiles and skews. It is perhaps the most popular method of managing exotic equity and foreign exchange options. Unfortunately, the local volatility model predicts the wrong dynamics of the implied volatility curve, which leads to inaccruate and often unstable hedges. To illustrate the problem, consider the special case in which the local vol is a function ofˆ? only: . (ˆ ?) to reproduce the option prices at ?1 . for all. Etcétera. This calibration process can be greatly simplified ??for all strikes. then calibrates ?(2) . (ˆ ?) to (2.7)?ˆ? =. (ˆ?)ˆ. ˆ?(0) =. 5
In [14] and [15] singular perturbation methods were used to analyze this model. There it was found that European call and put prices are given by Black’s formula 2.4a - 2.4c with the implied volatility ½ On the right hand side, the first term dominates the solution and the second term provides a much smaller correction The omitted terms are very small, usually less than 1% of the first term. The behavior of local volatility models can be largely understood by examining the first term in 2.8. The implied volatility depends on both the strike. and the current forward price. So supppose that today the forward price is ?0and the implied volatility curve seen in the marketplace is ?0 model to the market clearly requires choosing the local volatility to be (2.8) ??(. ) =. (1 2[? + ?])1 + 1 24 ?00 . (1 . (1 2[? + ?]) 2[? + ?])(? − ?)2+ ··· ? ?(?). Calibrating the (2.9) . (ˆ?) = ?0 ?(2ˆ? − ?0) ? Now that the model is calibrated, let us examine its predictions. Suppose that the forward value changes from ?0to some new value. From 2.8, 2.9 we see that the model predicts that the new implied volatility curve is (2.10) ??(. ) = ?0 ?(? +. − ?0) for an option with strike. given that the current value of the forward price is. In particular, if the forward price ?0 increases to. the implied volatility curve moves to the left; if ?0 decreases to. the implied volatility curve moves to the right. Local volatility models predict that the market smile/skew moves in the opposite direction as the price of the underlying asset. This is opposite to typical market behavior, in which smiles and skews move in the same direction as the underlying. To demonstrate the problem concretely, suppose that today’s implied volatility is a perfect smile (2.11a) ?0 ?(?) =. + ?[? − ?0]2 around today’s forward price ?0. Then equation 2.8 implies that the local volatility is (2.11b) . (ˆ?) =. + 3?(ˆ? − ?0)2+ ··· ? As the forward price. evolves away from ?0due to normal market fluctuations, equation 2.8 predicts that the implied volatility is (2.11c) ??(. ) =. + ?[? − (3 2?0−1 2?)]2+3 4?(? − ?0)2+ ··· ? . The implied volatility curve not only moves in the opposite direction as the underlying, but the curve also shifts upward regardless of whether. increases or decreases. Exact results are illustrated in figures 2.2 - 2.4. There we assumed that the local volatility. (ˆ?) was given by 2.11b, and used finite difference methods to obtain essentially exact values for the option prices, and thus implied volatilites. Hedges calculated from the local volatility model are wrong. To see this, let. (. ) be Black’s formula 2.4a - 2.4c for, say, a call option. Under the local volatility model, the value of a call option is given by Black’s formula (2.12a) . =. (. (. ). ) with the volatility. (. ) given by 2.8. Differentiating with respect to. yields the ∆ risk (2.12b)∆ ≡. ? =. ? +. . . (. ) ?? ? 6
0.18 0.20 0.22 0.24 0.26 0.28 8090 100 f0 110 120 Implied Vol K Fig. 2.2. Exact implied volatility. (. 0) (solid line) obtained from the local volatility. (ˆ?) (dashed line)? 0.18 0.20 0.22 0.24 0.26 0.28 80 90 f 100 f0 110 120 K Implied Vol Fig. 2.3. Implied volatility. (. ) if the forward price decreases from ?0to. (solid line). predicted by the local volatility model. The first term is clearly the ∆ risk one would calculate from Black’s model using the implied volatility from the market. The second term is the local volatility model’s correction to the ∆ risk, which consists of the Black vega risk multiplied by the predicted change in. due to changes in the underlying forward price. In real markets the implied volatily moves in the opposite direction as the direction predicted by the model. Therefore, the correction term needed for real markets should have the opposite sign as the correction predicted by the local volatility model. The original Black model yields more accurate hedges than the local volatility model, even though the local vol model is self-consistent across strikes and Black’s model is inconsistent. Local volatility models are also peculiar theoretically. Using any function for the local volatility. (??ˆ?) except for a power law, ?(??∗) = ?(?)ˆ. . (??ˆ?) = ?(?)ˆ. ˆ? = ?(?)?ˆ?1−?? (2.13) (2.14) introduces an intrinsic “length scale” for the forward priceˆ? into the model. That is, the model becomes inhomogeneous in the forward priceˆ. Although intrinsic length scales are theoretically possible, it is difficult to understand the financial origin and meaning of these scales [16], and one naturally wonders whether such 7
0.18 0.20 0.22 0.24 0.26 0.28 8090 100 f0 110 f 120 K Implied Vol Fig. 2.4. Implied volatility. (. ) if the forward prices increases from ?0 to. (solid line). scales should be introduced into a model without specific theoretical justification. 2.3. The SABR model. The failure of the local volatility model means that we cannot use a Markov - ian model based on a single Brownian motion to manage our smile risk. Instead of making the model non-Markovian, or basing it on non-Brownian motion, we choose to develop a two factor model. To select the second factor, we note that most markets experience both relatively quiescent and relatively chaotic periods. This suggests that volatility is not constant, but is itself a random function of time. Respecting the preceding discusion, we choose the unknown coefficient ?(??∗) to be ˆ? ˆ. where the “volatility” ˆ. is itself a stochastic process. Choosing the simplest reasonable process for ˆ. now yields the “stochastic-. model,” which has become known as the SABR model. In this model, the forward price and volatility are (2.15a)?ˆ? = ˆ? ˆ. 1? ˆ?(0) = ? (2.15b)?ˆ. =?ˆ. 2? ˆ ?(0) = ? under the forward measure, where the two processes are correlated by: (2.15c) ??1??2=. Many other stochastic volatility models have been proposed, for example [17], [18], [19], [20]; these models will be treated in section 5. However, the SABR model has the virtue of being the simplest stochastic volatility model which is homogenous inˆ? and ˆ. We shall find that the SABR model can be used to accurately fit the implied volatility curves observed in the marketplace for any single exercise date. More importantly, it predicts the correct dynamics of the implied volatility curves. This makes the SABR model an effective means to manage the smile risk in markets where each asset only has a single exercise date; these markets include the swaption and caplet/floorlet markets. As written, the SABR model may or may not fit the observed volatility surface of an asset which has European options at several different exercise dates; such markets include foreign exchange options and most equity options. Fitting volatility surfaces requires the dynamic SABR model which is introduced and analyzed in section 4. It has been claimed by many authors that stochastic volatility models are models of incomplete markets, because the stochastic volatility risk cannot be hedged. Esto no es verdad. It is true that the risk to changes in ˆ. (the vega risk) cannot be hedged by buying or selling the underlying asset. However, vega risk can be 8
hedged by buying or selling options on the asset in exactly the same way that ∆-hedging is used to neutralize the risks to changes in the priceˆ. In practice, vega risks are hedged by buying and selling options as a matter of routine, so whether the market would be complete if these risks were not hedged is a moot question. The SABR model 2.15a - 2.15c is analyzed in Appendix B. There singular perturbation techniques are used to obtain the prices of European options. From these prices, the options’ implied volatility. (. ) is then obtained. The upshot of this analysis is that under the SABR model, the price of European options is given by Black’s formula, . = ?(. ) ? . =. + ?(. )[? −. ]? (2.16a) (2.16b) with (2.16c) ?1?2=log. ±1 2?2 . ??√. ? where the implied volatility. (. ) is given by ??(. ) = ? (??)(1−?)?2n 1 + 1 +(1−?)2 24 log2. +(1−?)4 1920log4. + ··· (??)(1−?)?2+2−3?2 o · µ ? ?(?) ¶ · ½· (1−?)2 24 ?2 (??)1−?+1 4 . 24?2 ¸ . + ··· ? (2.17a) Here (2.17b) ? =? ?(??)(1−?)?2log. and ?(?) is defined by (2.17c) ?(?) = log (p1 − 2. + ?2+. − ? 1 − ? ) ? For the special case of at-the-money options, options struck at. =. this formula reduces to ½ These formulas are the main result of this paper. Although it appears formidable, the formula is explicit and only involves elementary trignometric functions. Implementing the SABR model for vanilla options is very easy, since once this formula is programmed, we just need to send the options to a Black pricer. In the next section we examine the qualitative behavior of this formula, and how it can be used to managing smile risk. The complexity of the formula is needed for accurate pricing. Omitting the last line of 2.17a, for example, can result in a relative error that exceeds three per cent in extreme cases. Although this error term seems small, it is large enough to be required for accurate pricing. The omitted terms “+···” are much, much smaller. Indeed, even though we have derived more accurate expressions by continuing the perturbation expansion to higher order, 2.17a - 2.17c is the formula we use to value and hedge our vanilla swaptions, caps, and floors. We have not implemented the higher order results, believing that the increased precision of the higher order results is superfluous. There are two special cases of note. = 1, representing a stochastic log normal model), and. = 0, representing a stochastic normal model. The implied volatility for these special cases is obtained in the last section of Appendix B. (2.18) . =. (. ) = ? ?(1−?) 1 + · (1−?)2 24 ?2 ?2−2?+1 4 . ?(1−?)+2−3?2 24?2 ¸ . + ··· ? 9
3. Managing smile risk. The complexity of the above formula for. (. ) obscures the qualitative behavior of the SABR model. To make the model’s phenomenology and dynamics more transparent, note that formula 2.17a - 2.17c can be approximated as ??(. ) = ? ?1−? ©1 −1 2(1 −. −. )log. +1 12 (3.1a) £(1 − ?)2+ (2 − 3?2)?2¤log2. + ··· ? provided that the strike. is not too far from the current forward. Here the ratio (3.1b) ? =? ??1−? measures the strength. of the volatility of volatility (the “volvol”) compared to the local volatility. 1−? at the current forward. Although equations 3.1a - 3.1b should not be used to price real deals, they are accurate enough to depict the qualitative behavior of the SABR model faithfully. Como. varies during normal trading, the curve that the ATM volatility. (. ) traces is known as the backbone, while the smile and skew refer to the implied volatility. (. ) as a function of strike. for a fixed. That is, the market smile/skew gives a snapshot of the market prices for different strikes. at a given instance, when the forward. has a specific price. Figures 3.1 and 3.2. show the dynamics of the smile/skew predicted by the SABR model. 8% 10% 12% 14% 16% 18% 20% 22% 4% 6% 8% 10%12% Imp . Higo. 3.1. Backbone and smiles for. = 0. As the forward. varies, the implied volatiliity. (. ) of ATM options traverses the backbone (dashed curve). Shown are the smiles. (. ) for three different values of the forward. Volatility data from 1 into 1 swaption on 4/28/00, courtesy of Cantor-Fitzgerald. Let us now consider the implied volatility. (. ) in detail. The first factor. 1−?in 3.1a is the implied volatility for at-the-money (ATM) options, options whose strike. equals the current forward. So the backbone traversed by ATM options is essentially. (. ) =. 1−?for the SABR model. The backbone is almost entirely determined by the exponent. with the exponent. = 0 (a stochastic Gaussian model) giving a steeply downward sloping backbone, and the exponent. = 1 giving a nearly flat backbone. The second term −1 respect to the strike. The −1 0≤. ≤ 1. It arises because the “local volatility” ˆ? ˆ. ˆ?1= ˆ. ˆ?1−?is a decreasing function of the forward price. The second part1 2??log. is the vanna skew, the skew caused by the correlation between 10 2(1 −. −. )log. represents the skew, the slope of the implied volatility with 2(1 − ?)log. part is the beta skew, which is downward sloping since
8% 10% 12% 14% 16% 18% 20% 22% 4%6% 8%10%12% Implie . Higo. 3.2. Backbone and smiles as above, but for. = 1. the volatility and the asset price. Typically the volatility and asset price are negatively correlated, so on average, the volatility. would decrease (increase) when the forward. increases (decreases). It thus seems unsurprising that a negative correlation. causes a downward sloping vanna skew. It is interesting to compare the skew to the slope of the backbone. Como. changes to ?0the ATM vol changes to (3.2a) ??(?0??0) = ? ?1−? ? + ···>? Near. =. el. component of skew expands as (3.2b) ??(. ) = ? ?1−? 2(1 − ?)? − ? ? + ···>? so the slope of the backbone. (. ) is twice as steep as the skew in. (. ) caused by. The last term in 3.1a also contains two parts. The first part (quadratic) term, but it is dominated by the downward sloping beta skew, and, at reasonable strikes. it just modifies this skew somewhat. The second part (vol-gamma) effect. Physically this smile arises because of “adverse selection”: unusually large movements of the forwardˆ? happen more often when the volatility. increases, and less often when. decreases, so strikes. far from the money represent, on average, high volatility environments. 1 12(1 − ?)2log2. appears to be a smile 1 12(2−3?2)?2log2. is the smile induced by the volga 3.1. Fitting market data. The exponent. and correlation. affect the volatility smile in similar ways. They both cause a downward sloping skew in. (. ) as the strike. varies. From a single market snapshot of. (. ) as a function of. at a given. it is difficult to distinguish between the two parameters. This is demonstrated by figure 3.3. There we fit the SABR parameters. con. = 0 and then re-fit the parameters. con. = 1. Note that there is no substantial difference in the quality of the fits, despite the presence of market noise. This matches our general experience: market smiles can be fit equally well with any specific value of. In particular. cannot be determined by fitting a market smile since this would clearly amount to “fitting the noise.” Figure 3.3 also exhibits a common data quality issue. Options with strikes. away from the current forward. trade less frequently than at-the-money and near-the-money options. Consequently, as. moves 11
1y into 1y 12% 14% 16% 18% 20% 22% 4%6%8%10% 12% Implie Fig. 3.3. Implied volatilities as a function of strike. Shown are the curves obtained by fitting the SABR model with exponent. = 0 and with. = 1 to the 1y into 1y swaption vol observed on 4/28/00. As usual, both fits are equally good. Data courtesy of Cantor-Fitzgerald. away from. the volatility quotes become more suspect because they are more likely to be out-of-date and not represent bona fide offers to buy or sell options. Suppose for the moment that the exponent. is known or has been selected. Taking a snapshot of the market yields the implied volatility. (. ) as a function of the strike. at the current forward price. Con. given, fitting the SABR model is a straightforward procedure. The three parameters. Y have different effects on the curve: the parameter. mainly controls the overall height of the curve, changing the correlation. controls the curve’s skew, and changing the vol of vol. controls how much smile the curve exhibits. Because of the widely seperated roles these parameters play, the fitted parameter values tend to be very stable, even in the presence of large amounts of market noise. The exponent. can be determined from historical observations of the “backbone” or selected from “aesthetic considerations.” Equation 2.18 shows that the implied volatility of ATM options is ½ The exponent. can be extracted from a loglog plot of historical observations of. pairs. Since both ? y. are stochastic variables, this fitting procedure can be quite noisy, and as the [···]. term is typically less than one or two per cent, it is usually ignored in fitting. Selecting. from “aesthetic” or other a priori considerations usually results in. = 1 (stochastic lognor - mal). = 0 (stochastic normal), or. =1 2(stochastic CIR) models. Proponents of. = 1 cite log normal models as being “more natural.” or believe that the horizontal backbone best represents their market. These proponents often include desks trading foreign exchange options. Proponents of. = 0 usually believe that a normal model, with its symmetric break-even points, is a more effective tool for managing risks, and would claim that. = 0 is essential for trading markets like Yen interest rates, where the forwards. can be nega - tive or near zero. Proponents of. =1 2are usually US interest rate desks that have developed trust in CIR 12 (3.3)log??(. ) = log? − (1 − ?)log? + log 1 + [(1−?)2 24 ?2 ?2−2?+1 4 . ?(1−?)+2−3?2 24?2]. + ··· ? ¾
models. It is usually more convenient to use the at-the-money volatility. y. as the SABR parame - ters instead of the original parameters. The parameter. is then found whenever needed by inverting 2.18 on the fly; this inversion is numerically easy since the [···]. term is small. With this parameterization, fitting the SABR model requires fitting. y. to the implied volatility curve, with. y. dado. In many markets, the ATM volatilities need to be updated frequently, say once or twice a day, while the smiles and skews need to be updated infrequently, say once or twice a month. With the new parameterization, . can be updated as often as needed, with. (and ?) updated only as needed. Let us apply SABR to options on US dollar interest rates. There are three key groups of European options on US rates: Eurodollar future options, caps/floors, and European swaptions. Eurodollar future options are exchange-traded options on the 3 month Libor rate; like interest rate futures, EDF options are quoted on 100(1−. ). Figure 1.1 fits the SABR model (with. = 1) to the implied volatility for the June 99 contracts, and figures 3.4 - 3.7 fit the model (also with. = 1) to the implied volatility for the September 99, December 99, and March 00 contracts. All prices were obtained from Bloomberg Information Services on March 23, 1999. Two points are shown for the same strike where there are quotes for both puts and calls. Note that market liquidity dries up for the later contracts, and for strikes that are too far from the money. Consequently, more market noise is seen for these options. U99 Eu rodo lla r op tio n 10 15 20 25 30 93.0 94.095.0 Strik e 96.0 97.0 Vol (%) Fig. 3.4. Volatility of the Sep 99 EDF options Caps and floors are sums of caplets and floorlets; each caplet and floorlet is a European option on the 3 month Libor rate. We do not consider the cap/floor market here because the broker-quoted cap prices must be “stripped” to obtain the caplet volatilities before SABR can be applied. A m year into n year swaption is a European option with m years to the exercise date (the maturity); if it is exercised, then one receives an n year swap (the tenor, or underlying) on the 3 month Libor rate. See Appendix A. For almost all maturities and tenors, the US swaption market is liquid for at-the-money swaptions, but is ill-liquid for swaptions struck away from the money. Hence, market data is somewhat suspect for swaptions that are not struck near the money. Figures 3.8 - 3.11 fits the SABR model (with ? = 1) to the prices of. into5? swaptions observed on April 28, 2000. Data supplied courtesy of Cantor - Fitzgerald. We observe that the smile and skew depend heavily on the time-to-exercise for Eurodollar future options and swaptions. The smile is pronounced for short-dated options and flattens for longer dated options; the skew is overwhelmed by the smile for short-dated options, but is important for long-dated options. This 13
Z99 Eurodolla r op tion 14 16 18 20 92.093.0 94.0 Strik e 95.0 96.0 Vol (%) Fig. 3.5. Volatility of the Dec 99 EDF options H00 Eu ro dolla r option 14 16 18 20 22 24 92.0 93.0 94.095.0 96.0 97.0 Strik e Vol (%) Fig. 3.6. Volatility of the Mar 00 EDF options picture is confirmed tables 3.1 and 3.2. These tables show the values of the vol of vol. and correlation ? obtained by fitting the smile and skew of each “m into n” swaption, again using the data from April 28, 2000. Note that the vol of vol. is very high for short dated options, and decreases as the time-to-exercise increases, while the correlations starts near zero and becomes substantially negative. Also note that there is little dependence of the market skew/smile on the length of the underlying swap; both. y. are fairly constant across each row. This matches our general experience: in most markets there is a strong smile for short-dated options which relaxes as the time-to-expiry increases; consequently the volatility of volatility ? is large for short dated options and smaller for long-dated options, regardless of the particular underlying. Our experience with correlations is less clear: in some markets a nearly flat skew for short maturity options develops into a strongly downward sloping skew for longer maturities. In other markets there is a strong downward skew for all option maturities, and in still other markets the skew is close to zero for all maturities. 14
M 00 Euro dolla r optio n 14 16 18 20 22 24 92.093.0 94.095.096.0 97.0 Strik e Vol (%) Fig. 3.7. Volatility of the Jun 00 EDF options 3M into 5Y 12% 14% 16% 18% 20% 4%6%8% 10% 12% Fig. 3.8. Volatilities of 3 month into 5 year swaption 3.2. Managing smile risk. After choosing. and fitting. and either. o. the SABR model (3.4a)?ˆ? = ˆ? ˆ. 1? ˆ?(0) = ? (3.4b)?ˆ. =?ˆ. 2? ˆ ?(0) = ? with (3.4c) ??1??2=. fits the smiles and skews observed in the market quite well, especially considering the quality of price quotes away from the money. Let us take for granted that it fits well enough. Then we have a single, self-consistent model that fits the option prices for all strikes. without “adjustment,” so we can use this model to price exotic options without ambiguity. The SABR model also predicts that whenever the forward price. changes, 15
1Y into 5Y 13% 14% 15% 16% 17% 18% 4%6% 8%10%12% Fig. 3.9. Volatilities of 1 year into 1 year swaptions 5Y into 5Y 12% 13% 14% 15% 16% 17% 4%6% 8% 10%12% Fig. 3.10. Volatilities of 5 year into 5 year swaptions the the implied volatility curve shifts in the same direction and by the same amount as the price. This predicted dynamics of the smile matches market experience. Si. 1, the “backbone” is downward sloping, so the shift in the implied volatility curve is not purely horizontal. Instead, this curve shifts up and down as the at-the-money point traverses the backbone. Our experience suggests that the parameters. Y are very stable (? is assumed to be a given constant), and need to be re-fit only every few weeks. This stability may be because the SABR model reproduces the usual dynamics of smiles and skews. In contrast, the at-the-money volatility. or, equivalently. may need to be updated every few hours in fast-paced markets. Since the SABR model is a single self-consistent model for all strikes. the risks calculated at one strike are consistent with the risks calculated at other strikes. Therefore the risks of all the options on the same asset can be added together, and only the residual risk needs to be hedged. Let us set aside the ∆ risk for the moment, and calculate the other risks. Let. (. ) be 16
10Y into 5Y 9% 10% 11% 12% 13% 4%6%8% 10%12% Fig. 3.11. Volatilities of 10 year into 5 year options ? 1Y2Y 3Y4Y5Y 7Y10Y 74.1% 59.5% 50.0% 44.7% 37.6% 32.2% 27.0% 24.5% 1M 3M 6M 1Y 3Y 5Y 7Y 10Y 76.2% 65.1% 57.1% 59.8% 42.1% 33.4% 30.2% 26.7% 75.4% 62.0% 52.6% 49.3% 39.1% 33.2% 29.2% 26.3% 74.6% 60.7% 51.4% 47.1% 38.4% 33.1% 29.0% 26.0% 74.1% 60.1% 50.8% 46.7% 38.4% 32.6% 28.2% 25.6% 75.2% 62.9% 49.4% 46.0% 36.9% 31.3% 26.2% 24.8% 73.7% 59.7% 50.4% 45.6% 38.0% 32.3% 27.2% 24.7% Table 3.1 Volatility of volatility. for European swaptions. Rows are time-to-exercise; columns are tenor of the underlying swap. Black’s formula 2.4a - 2.4c for, say, a call option. According to the SABR model, the value of a call is (3.5) . =. (. (. ). ) where the volatility. (. ) ≡. (. ;. ) is given by equations 2.17a - 2.17c. Differentiating1with respect to. yields the vega risk, the risk to overall changes in volatility: (3.6) . ? =. . ·. (. ;. ) ?? ? This risk is the change in value when. changes by a unit amount. It is traditional to scale vega so that it represents the change in value when the ATM volatility changes by a unit amount. Ya que. = (. ). the vega risk is (3.7a)vega ≡. ? =. . · . (. ;. ) ?? . (?;. ) ?? 1In practice risks are calculated by finite differences: valuing the option at. re-valuing the option after bumping. a ? +. and then subtracting to determine the risk. This saves differentiating complex formulas such as 2.17a - 2.17c. 17
1Y 4.2% 2.5% 5.0% -4.4% -7.3% -11.1% -13.7% -14.8% 2Y -0.2% -4.9% -3.6% -8.1% -14.3% -17.3% -22.0% -25.5% 3Y -0.7% -5.9% -4.9% -8.8% -17.1% -18.5% -23.6% -27.7% 4Y -1.0% -6.5% -5.6% -9.3% -17.1% -18.8% -24.0% -29.2% 5Y -2.5% -6.9% -7.1% -9.8% -16.6% -19.0% -25.0% -31.7% 7Y -1.8% -7.6% -7.0% -10.2% -17.9% -20.0% -26.1% -32.3% 10Y -2.3% -8.5% -8.0% -10.9% -18.9% -21.6% -28.7% -33.7% 1M 3M 6M 1Y 3Y 5Y 7Y 10Y Table 3.2 Matrix of correlations. between the underlying and the volatility for European swaptons. dónde. (?) =. (. ) is given by 2.18. Note that to leading order. ≈. y. ≈ . so the vega risk is roughly given by (3.7b)vega ≈. . ·??(. ) . (?)=. . ·??(. ) ??(. )? Qualitatively, then, vega risks at different strikes are calculated by bumping the implied volatility at each strike. by an amount that is proportional to the implied volatiity. (. ) at that strike. That is, in using equation 3.7a, we are essentially using proportional, and not parallel, shifts of the volatility curve to calculate the total vega risk of a book of options. Ya que. y. are determined by fitting the implied volatility curve observed in the marketplace, the SABR model has risks to. y. changing. Borrowing terminology from foreign exchange desks, vanna is the risk to. changing and volga (vol gamma) is the risk to. changing: (3.8a) vanna =. ? =. . ·. (. ;. ) ?? ? (3.8b) volga =. ? =. . ·. (. ;. ) ?? ? Vanna basically expresses the risk to the skew increasing, and volga expresses the risk to the smile becoming more pronounced. These risks are easily calculated by using finite differences on the formula for. in equations 2.17a - 2.17c. If desired, these risks can be hedged by buying or selling away-from-the-money options. The delta risk expressed by the SABR model depends on whether one uses the parameterization. . o. Suppose first we use the parameterization. so that. (. ) ≡ ??(. ;. ). Differentiating respect to. yields the ∆ risk (3.9)∆ ≡. ? =. ? +. . . (. ;. ) ?? ? The first term is the ordinary ∆ risk one would calculate from Black’s model. The second term is the SABR model’s correction to the ∆ risk. It consists of the Black vega times the predicted change in the implied volatility. caused by the change in the forward. As discussed above, the predicted change consists of a sideways movement of the volatility curve in the same direction (and by the same amount) as the change in the forward price. In addition, if. 1 the volatility curve rises and falls as the at-the-money point 18
traverses up and down the backbone. There may also be minor changes to the shape of the skew/smile due to changes in. Now suppose we use the parameterization. Entonces. is a function of. y. defined implicitly by 2.18. Differentiating 3.5 now yields the ∆ risk ½. (. ;. ) ?? The delta risk is now the risk to changes in. with. held fixed. The last term is just the change in ? needed to keep. constant while. Cambios. Clearly this last term must just cancel out the vertical component of the backbone, leaving only the sideways movement of the implied volatilty curve. Note that this term is zero for. = 1. Theoretically one should use the ∆ from equation 3.9 to risk manage option books. In many markets, however, it may take several days for volatilities. to change following significant changes in the forward price. In these markets, using ∆ from 3.10 is a much more effective hedge. For suppose one used ∆ from equation 3.9. Then, when the volatility. did not immediately change following a change in. one would be forced to re-mark. to compensate, and this re-marking would change the ∆ hedges. Como. equilibrated over the next few days, one would mark. back to its original value, which would change the ∆ hedges back to their original value. This “hedging chatter” caused by market delays can prove to be costly. (3.10)∆ ≡. ? +. . +. (. ;. ) ?? ??(. ) ?? ¾ ? 4. The dynamic SABR model. Quote results for smile and skew. For each exercise date, same smile as in the static SABR model! Same smile dynamics! Calibrating volatility surface is no harder than calibrating smile. Show some results. FX options? Appendix A. Martingale pricing. Martingale pricing theory [7], [6], [8], [27] envisions an economy in which all asset prices can be generated by a finite set of Brownian motions; in which all asset prices are non-anticipating; and in which all asset prices are regular enough to be written as Ito processes. Along with some technical assumptions (the “usual conditions”), these postulates lead to the fundamental theorem of arbitrage free finance: Theorem A.1. Let the numeraire N(t) be the value of any freely tradeable, positively valued asset which generates no cash flow. In the absence of arbitrage, there exists a probability measure. (which depends on the security chosen as the numeraire) such that for all other securities, the value V(t) of the security divided by the numeraire N(t) is a Martingale. Specifically, if V(t) generates no cash flow, then at any time t, ½? (?) (A.1a) ? (?) ?(?)=. ?(?)|F? ¾ for all. where the expected value. is over this probability measure. If V(t) generates cash flow C(t), then ( ? (A.1b) ? (?) ?(?)=. ? (?) ?(?)+ Z? ?(?0) ?(?0)??0|F? ) for all. In this appendix, we apply use this theorem to develop the theoretical framework for pricing European options and swaptions. A.1. General European options. Consider a financial asset A? Suppose we have a forward contact on this asset which requires us to pay the strike price. and acquire the asset on the settlement date. Letˆ ?(?) be the asset’s current price, let. (?) be the value of the forward contract, and let ?(?;. ) be the value of a zero coupon bond which pays $1 on the settlement date. 19
To price the forward contract, let us choose the zero coupon bond for our numeraire. Then there exists a probability measure. (known as the forward measure), such that. (?)??(?;. ) is a Martingale: ½ At the settlement date, clearly. (. ) =ˆ ?(. )−? and ?(. ;. ) = 1, so evaluating this expression at ? =. yields h Clearly ˆ?(?) =. nˆ ?(. )|F? is the forward price of the asset for date. as seen at date. since it is the strike at which the forward contract would have no value on date. The value of the forward contract can now be written as (A.2) . (?) ?(. )=. . (?) ?(?;. )|F? ¾ for all. (A.3) . (?) = ?(?;. ) ??nˆ ?(. )|F? o − ? i ? (A.4) o (A.5) . (?) = ?(?;. ) hˆ?(?) − ? i ? the usual formula for a forward contract. Observe that the forward price is a Martingale in the forward measure; this can be seen from the “telescoping" rule of expected values. Alternatively, one can simply note that since ?(?;. ) is our numeraire, and a forward contract is certainly a tradeable security, the fundamental theorem shows that. (?)??(?;. ), and thusˆ?(?) −. is a Martingale. Ya que. is a constant, clearlyˆ?(?) is a Martingale. The Martingale representation theorem now shows that (A.6)?ˆ? = ?(??∗). for some coefficient ?(??∗), where. is Brownian motion under. Now consider a European call option on the same asset A? If one exercises a European option on its exercise date. then the asset will be acquired, and the strike paid, on the settlement date. of the option. That is, when one exercises a European option on. one usually receives a forward contract on the asset, and not immediate possesion of the asset. To price the European call option, let us again choose the zero coupon bond ?(?;. ) as the numeraire. The value of the option is ½. (?) The simplest date. to use is the exercise date. Since the option will be exercised if the forward contract has positive value, the value of the option at date. =. is clearly [. (. ]+, which is ?(. ;. ) Substituting this into the above equation yields (A.7) . (?) ?(. )=. ?(?;. )|F? ¾ for all. hˆ?(. ) − ? i+. (A.8a) . (?) = ?(. )?? ½hˆ?(. ) − ? i+ |F? ¾ for the value of the European option, where the expectation is over the forward measure. Moreover, in this measure, the forward price is a Martingale, (A.8b) . = ?(??∗). Letting today be. = 0, and noting that today’s values of zero coupon bonds ?(0. ) are just the discount factors ?(. ), now reduces A.8a, A.8b to equations 2.1a, 2.1b, the starting point of our analysis.. 20
A.2. European swaptions. European swaptions fit naturally into the same framework. A standard swap is defined by specifying the fixed rate (strike). the start date ?0, and the theoretical end date˜. In most currencies, the default frequency for the fixed leg is semiannual, and the theoretical dates˜?1?˜?2. ˜?? are generated by subtracting 6 months, 12 months, 18 months. from the theoretical end date˜. The actual dates ?1??2. are then obtained by replacing any bad business days according to the modified following business day convention. Once the actual dates are constructed, the fixed leg pays . paid at. for. = 1?2. − 1? (A.9a) 1 +. paid at. (A.9b) Here the coefficient. = ?(??−1. ) is the year fraction between. −1and. Year fractions for semiannual legs are often not exactly1 2due to the vagaries of holidays, weekends, and shorter or longer months. They are determined by the swap’s day count basis, which is typically 30/360 for USD legs, actual/365 for Sterling legs, and 30/360E for Euro legs. The final payment includes repayment of the notional, which is assumed to be 1. On any date. the value of the fixed leg is clearly the same as a collection of zero coupon bonds: (A.10) . (?) =. ? X ?=1 . (?;??) + ?(??)? The actual dates ?1??2. =. of the floating leg are set up in exactly the same way, save that floating legs are usually quarterly instead of semiannual. The floating rate pays . ? paid at. for. = 1?2. − 1? (A.11a) 1 +. ? paid at. (A.11b) where. is the day count fraction of the. interval according to the floating leg’s day count basis. The interest rate. ? para el. interval. −1. ≤. is set spot-lag (generally 2) business days before the interval starts at. −1? Theoretically, the value of a floating leg is the same as the value of 1 paid at the start date ?0of the floating leg: (A.12) . (?) = ?(?;?0)? This, because if one starts with a single unit at ?0, one can invest it at the rate. then reinvest 1 at the new rate. 2from ?1to ?2. and finally reinvest it at. a. Thus, starting with 1 unit of currency enables one to reconstruct the floating leg payments exactly. A (payer) swap exchanges a fixed leg for a floating leg. Its value is 1from ?0until ?1, one can? for the final interval. −1 (A.13) . (?) = ?(?;?0) − ?(?;??) −. ? X ?=1 . (?;??)? The swap rate is defined to be the value of. which sets the value of this swap to 0. So at any date. the swap rate is (A.14a) ??(?) =?(?0) − ?(??) ?(?) ? 21
where (A.14b) ?(?) = ? X ?=1 . (?;??) is the level (also called the annuity or PV01). The value of the swap can now be written in terms of the swap rate as (A.14c) . (?) = [??(?) −. ]?(?)? Consider a swaption, a European option on the swap. To price the option, we note that the level is clearly a tradeable instrument, since it is just a collection of zero coupon bonds, so we choose it as our numeraire. The fundamental theorem ensures that there is a probability measure. known as the level measure, such that for any tradeable instrument, the instrument’s value. (?) divided by ?(?) is a Martingale. In particular, the value of the swaption is ½. (?) The simplest date. for evaluating this expression is the swaption’s exercise date. On the exercise date, the value of the swaption is clearly [. (. )]+, which is [??(. ) −. ]+?(. )? Therefore the swaption value can be written as . (?) = ?(?)??n where the expectation is over the level measure. Moreover, since ?(?0)−?(??) represents a tradeable security (long a zero coupon bond, and short a second zero coupon bond), and ?(?) is our numeraire, the swap rate ??(?) is a Martingale in this measure. So from the Martingale representation theorem, we again conclude that (A.15) . (?) ?(?) =. ?(?) |F? ¾ for all. (A.16) [??(. ) −. ]+|F? o (A.17) . = ?(??∗). for some coefficient. (??∗)? dónde. is Brownian motion. Finally, letting. = 0 be today, and and noting that today’s zero coupon bond values ?(0;??) are just the discount factors ?(??), we can write today’s swaption value as (A.18a) . (0) = ?0??n [??(. ) −. ]+|F0 o ? where the swap rate. (?) is a Martingale, (A.18b) . = ?(??∗). and today’s level ?0and today’s swap rate are given by (A.18c) ?0= ? X ?=1 . (??). (0) =?(?0) − ?(??) ?0 ? This is clearly the same theoretical framework as the general European option above. Appendix B. Analysis of the SABR model. 22
Here we use singular perturbation techniques to price European options under the SABR model. Our analysis is based on a small volatility expansion, where we take both the volatility ˆ. and the “volvol”. to be small. To carry out this analysis in a systematic fashion, we re-write ˆ. −→?ˆ. y. −→. and analyze? ˆ? =?ˆ. (ˆ?)??1? (B.1a)?ˆ. =. ˆ. 2? (B.1b) with (B.1c) ??1??2=. in the limit. ¿ 1. This is the distinguished limit [22], [23] in the language of singular perturbation theory. After obtaining the results we replace? ˆ. −→ ˆ. y. −→. to get the answer in terms of the original variables. We first analyze the model with a general ?(ˆ?), and then specialize the results to the power law ˆ. This is notationally simpler than working with the power law throughout, and the more general result may prove valuable in some future application. We first use the forward Kolmogorov equation to simplify the option pricing problem. Suppose the economy is in stateˆ?(?) =. ˆ ?(?) =. at date. Define the probability density ?(. ;. ) by n As a function of the forward variables. the density. satisfies the forward Kolmogorov equation (the F˝ okker-Planck equation) (B.2) ?(. ;. ). = prob . ˆ?(?). +. ˆ ?(?). +. ¯¯¯ˆ?(?) =. ˆ ?(?) = ? O (B.3a) ??=1 2?2?2[?2(?)?]??+ ?2??[?2?(?)?]??+1 2?2?2[?2?]?? for. with (B.3b) ? = ?(? − ?)?(? − ?) at. =. as is well-known [25], [26], [27]. Here, and throughout, we use subscripts to denote partial derivatives. Let. (. ) be the value of a European call option at date. when the economy is in stateˆ?(?) = . ˆ ?(?) =. Let. be the option’s exercise date, and let. be its strike. Omitting the discount factor ?(. ), which factors out exactly, the value of the option is n = −∞ ? ? (. ) = ? [ˆ?(. ) − ?]+|ˆ?(?) =. ˆ ?(?) = ? Z∞ o (B.4) Z∞ (? − ?)?(. ;. ). See 2.1a. Since (B.5) ?(. ;. ) = ?(? − ?)?(? − ?) + Z. ? ??(. ;. ). we can re-write. (. ) as (B.6) ? (. ) = [? − ?]++ Z. ? Z∞ ? Z∞ −∞ 23 (? − ?)??(. ;. ).
We substitute B.3a for. into B.6. Integrating the. derivatives ?2??[?2?(?)?]??and1 all. yields zero. Therefore our option price reduces to Z. ? where we have switched the order of integration. Integrating by parts twice with respect to. now yields 2?2?2[?2?]??over (B.7) ? (. ) = [? − ?]++1 2?2 Z∞ −∞ Z∞ ? ?2(? − ?)[?2(?)?]. (B.8) ? (. ) = [? − ?]++1 2?2?2(?) Z. ? Z∞ −∞ ?2?(. ;. ). The problem can be simplified further by defining (B.9) ?(. ;. ) = Z∞ −∞ ?2?(. ;. ). Entonces. satisfies the backward’s Kolmogorov equation [25], [26], [27] ??+1 2?2?2?2(?). + ?2. 2?(?). +1 2?2?2?2. = 0? ? = ?2?(? − ?)? for. (B.10a) for. =. (B.10b) Since. does not appear explicitly in this equation. depends only on the combination. −. and not on ? y. separately. So define (B.11) ? =. −. =. −. Then our pricing formula becomes (B.12) ? (. ) = [? − ?]++1 2?2?2(?) Z. 0 ?(. ;?)?? where ?(. ;?) is the solution of the problem ??=1 ? = ?2?(? − ?)? 2?2?2?2(?). + ?2. 2?(?). +1 for. = 0? 2?2?2?2. for. 0? (B.13a) (B.13b) In this appendix we solve B.13a, B.13b to obtain ?(. ;?), and then substitute this solution into B.12 to obtain the option value. (. ). This yields the option price under the SABR model, but the resulting formulas are awkward and not very useful. To cast the results in a more usable form, we re-compute the option price under the normal model (B.14a)?ˆ? =. and then equate the two prices to determine which normal volatility. needs to be used to reproduce the option’s price under the SABR model. That is, we find the “implied normal volatility” of the option under the SABR model. By doing a second comparison between option prices under the log normal model (B.14b)?ˆ? =. ˆ. and the normal model, we then convert the implied normal volatility to the usual implied log-normal (Black - Scholes) volatility. That is, we quote the option price predicted by the SABR model in terms of the option’s implied volatility. 24
B.1. Singular perturbation expansion. Using a straightforward perturbation expansion would yield a Gaussian density to leading order, (B.15a) ? = ? p2??2?2(?)??− (?−?)2 2?2?2?2(?)? ? Since the “+···” involves powers of (? − ?). (?), this expansion would become inaccurate as soon as (? − ?)?0(?)??(?) becomes a significant fraction of 1; i. e. as soon as ?(?) and ?(?) are significantly different. Stated differently, small changes in the exponent cause much greater changes in the probability density. A better approach is to re-cast the series as (B.15b) ? = ? p2??2?2(?)??− (?−?)2 2?2?2?2(?)? and expand the exponent, since one expects that only small changes to the exponent will be needed to effect the much larger changes in the density. This expansion also describes the basic physics better –. is essentially a Gaussian probability density which tails off faster or slower depending on whether the “diffusion coefficient” ?(?) decreases or increases. We can refine this approach by noting that the exponent is the integral (B.16) (? − ?)2 2?2?2?2(?)? = 1 2? Ã 1 ?? Z? ? ??0 ?(?0) !2 ? Suppose we define the new variable (B.17) ? = 1 ?? Z? ? ??0 ?(?0)? so that the solution. is essentially ?−?2?2. To leading order, the density is Gaussian in the variable. which is determined by how “easy” or “hard” it is to diffuse from. a. which closely matches the underlying physics. The fact that the Gaussian changes by orders of magnitude as ?2increases should be largely irrelevent to the quality of the expansion. This approach is directly related to the geometric optics technique that is so successful in wave propagation and quantum electronics [28], [23]. To be more specific, we shall use the near identity transform method to carry out the geometric optics expansion. This method, pioneered in [29], transforms the problem order-by-order into a simple canonical problem, which can then be solved trivially. Here we obtain the solution only through ?(?2), truncating all higher order terms. Let us change variables from. to (B.18a) ? = 1 ?? Z? ? ??0 ?(?0)? and to avoid confusion, we define (B.18b) ?(. ) = ?(?)? Then (B.19a) ? ??−→ 1 . (?) ? ??= 1 . (. ) ? . ? ??−→ ? ??−? ? ? . 25
There are no longer any. derivatives, so we can now treat. as a parameter instead of as an independent variable. That is, we have succeeded in effectively reducing the problem to one dimension. Let us now remove the. term through ?(?2). To leading order. 0(. )??(. ) and ?00(. )??(. ) are constant. We can replace these ratios by (B.32) ?1= ?0(. 0)??(. 0)??2= ?00(. 0)??(. 0)? commiting only an ?(?) error, where the constant ?0will be chosen later. We now defineˆ. by (B.33) ? =. 2. 1?2?4ˆ. Then our option price becomes (B.34) ? (. ) = [? − ?]++1 2??p?(0)?(. )??2. 1?2?4 Z. 0 ˆ?(. ). whereˆ. is the solution of (B.35a) ˆ. =1 2 ¡1 − 2. + ?2?2?2¢ˆ. + ?2?2¡1 ˆ. = ?(?) at. = 0? 4?2−3 8?2 1 ¢ˆ. +3 4?2. 1ˆ ? for. 0 (B.35b) We’ve almost beaten the equation into shape. We now define (B.36a) ? = 1 ?? Z. 0 ?? p 1 − 2. + ?2= 1 ??log( p1 − 2. + ?2?2?2−. +. 1 − ? )? which can be written implicitly as (B.36b) . = sinh. − ?(cosh. − 1)? In terms of x, our problem is (B.37) ? (. ) = [? − ?]++1 2??p?(0)?(. )??2. 1?2?4 Z. 0 ˆ ?(. ). with (B.38a) ˆ. =1 2ˆ. −1 2. 0(. )ˆ. + ?2?2¡1 at. = 0? 4?2−3 8?2 1 ¢ˆ. +3 4?2. 1ˆ ? for. 0 (B.38b) ˆ? = ?(?) Here (B.39) ?(?) = q 1 − 2. + ?2? The final step is to define. by (B.40) ˆ. = ?1?2(. (?))? =¡1 − 2. + ?2?2?2¢1?4?? 28
through ?(?2). This can be shown by expanding ?2? through ?(?2), and noting that ?2. 2matches. 3. Therefore our option price is Z. (B.49) ? (. ) = [? − ?]++1 2 ? − ? ? 0 1 √2. −?2?2. 2? ? µ 1 −2? ?2?2? ¶3?2? and changing integration variables to (B.50) ? =?2 2?? reduces this to (B.51) ? (. ) = [? − ?]++|? − ?| 4√? Z∞ 2. ?2 ?−?+?2? (? − ?2?)3?2. That is, the value of a European call option is given by (B.52a) ? (. ) = [? − ?]++|? − ?| 4√? Z∞ ?2 2. −?2? ?−? ?3?2. with (B.52b) ?2? = log µ . ? − ? p?(0)?(. ) ¶ + log µ??1?2(. ) ? ¶ +1 4?2. 1?2? through ?(?2). B.2. Equivalent normal volatility. Equations B.52a and B.52a are a formula for the dollar price of the call option under the SABR model. The utility and beauty of this formula is not overwhelmingly apparent. To obtain a useful formula, we convert this dollar price into the equivalent implied volatilities. We first obtain the implied normal volatility, and then the standard log normal (Black) volatility. Suppose we repeated the above analysis for the ordinary normal model (B.53a)?ˆ? =. ˆ?(0) =. where the normal volatily. is constant, not stochastic. (This model is also called the absolute or Gaussian model). We would find that the option value for the normal model is exactly (B.53b) ? (. ) = [? − ?]++|? − ?| 4√? Z∞ 2?2 (?−?)2 . ?−? ?3?2?? This can be seen by setting ?(?) to 1, setting. a. and setting. to 0 in B.52a, B.52b. Working out this integral then yields the exact European option price (B.54a) ? (. ) = (? − ?)N(? − ? ??√. ) +. √. G(? − ? ??√. )? for the normal model, where N is the normal distribution and G is the Gaussian density (B.54b)G(?) = 1 √2??−?2?2? 30
From B.53b it is clear that the option price under the normal model matches the option price under the SABR model B.52a, B.52a if and only if we choose the normal volatility. to be ½ Taking the square root now shows the option’s implied normal (absolute) volatility is given by ½ through ?(?2). Before continuing to the implied log normal volatility, let us seek the simplest possible way to re-write this answer which is correct through ?(?2). Ya que. = ?[1 + ?(?)], we can re-write the answer as µ? − ? ? where à (B.55) 1 ?2 ? = ?2 (? − ?)2 1 − 2?2? ?2. ¾ ? (B.56) ??=? − ? ? 1 + ?2? ?2. + ··· ¾ (B.57a) ??= ¶µ ? ?(?) ¶©1 + ?2(?1+ ?2+ ?3). + ···ª? ? − ? ? =??(? − ?) R? ? ??0 ?(?0) = 1 ? − ? Z? ? ??0 . (?0) !−1 ? This factor represents the average difficulty in diffusing from today’s forward. to the strike. and would be present even if the volatility were not stochastic. The next factor is (B.57b) ? ?(?)= ? log Ãp 1 − 2. + ?2−. + ? 1 − ? Unesdoc. unesco. org unesdoc. unesco. org where (B.57c) ? =. =? ? Z? ? ??0 ?(?0)=? ? ? − ? ?(. ) ©1 + ?(?2)ª? Aquí. =√. is the geometric average of. y. (The arithmetic average could have been used equally well at this order of accuracy). This factor represents the main effect of the stochastic volatility. The coefficients ?1. 2, and ?3provide relatively minor corrections. Through ?(?2) these corrections are µ ?2?2= ?2log ? 4?2. 0(. 0) ?(. 0)=1 ?2?1= 1 ?2log . ? − ? ³? p?(?)?(?) £1 − 2. + ?2?2?2¤1?4´ 4?2. 1?(. ) + ··· ¶ =2?2− ?2 1 24 =2 − 3?2 ?2?2?2(. ) + ···(B.57d) 1 24 ?2?2+ ···(B.57e) ?2?3=1 (B.57f) where (B.57g) ?1=?0(. ) ?(. )??2=?00(. ) ?(. )? 31
Let us briefly summarize before continuing. Under the normal model, the value of a European call option with strike. and exercise date. is given by B.54a, B.54b. For the SABR model,?ˆ? =?ˆ. (ˆ?)??1??ˆ. =. ˆ. 2? ˆ?(0) = ? (B.58a) ˆ ?(0) = ? (B.58b) (B.58c) ??1??2=. the value of the call option is given by the same formula, at least through ?(?2), provided we use the implied normal volatility µ ½ Here ??(?) =??(? − ?) R? ? ??0 ?(?0) · ? ˆ ?(?) ¶ · (B.59a) 1 + ·2?2− ?2 1 24 ?2?2(. ) +1 4. 1? (. ) +2 − 3?2 24 ?2 ¸ ?2. + ··· ¾ ? (B.59b) . = p. 1=?0(. ) ?(. )??2=?00(. ) ?(. )? (B.59c) ? =? ? ? − ? ?(. )? ˆ ?(?) = log Ãp 1 − 2. + ?2−. + ? 1 − ? ! ? The first two factors provide the dominant behavior, with the remaining factor 1+[···]?2. usually provide - ing corrections of around 1% or so. One can repeat the analysis for a European put option, or simply use call/put parity. This shows that the value of the put option under the SABR model is (B.60) . (. ) = (? − ?)N(? − ? ??√. ) +. √. G(? − ? ??√. ) where the implied normal volatility. is given by the same formulas B.59a - B.59c as the call. We can revert to the original units by replacing. −→. −→. everywhere in the above formulas; this is equivalent to setting. to 1 everywhere. B.3. Equivalent Black vol. With the exception of JPY traders, most traders prefer to quote prices in terms of Black (log normal) volatilities, rather than normal volatilities. To derive the implied Black volatility, consider Black’s model (B.61)?ˆ? =. ˆ. ˆ?(0) =. where we have written the volatility as. to stay consistent with the preceding analysis. For Black’s model, the value of a European call with strike. and exercise date. is . =?N(?1) −?N(?2)? . =. + ?(. )[? − ?]? 32 (B.62a) (B.62b)
with (B.62c) ?1?2=log. ±1 2?2?2 . . √. ? where we are omitting the overall factor ?(. ) as before. We can obtain the implied normal volatility for Black’s model by repeating the preceding analysis for the SABR model with ?(?) =. y. = 0. Setting ?(?) =. y. = 0 in B.59a - B.59c shows that the normal volatility is (B.63) ??(?) =. (? − ?) log. ©1 − 1 24?2?2 . + ···ª? through ?(?2). Indeed, in [15] it is shown that the implied normal volatility for Black’s model is (B.64) ??(?) =. p ?? 1 + 1 24log2. + 1 1 1920log4. + ··· . + 1 + 1 24 ¡1 − 120log2. ¢?2?2 1 5760?4?4 ??2 ??+ ··· through ?(?4). We can find the implied Black vol for the SABR model by setting. obtained from Black’s model in equation B.63 equal to. obtained from the SABR model in B.59a - B.59c. Through ?(?2) this yields µ ½ This is the main result of this article. As before, the implied log normal volatility for puts is the same as for calls, and this formula can be re-cast in terms of the original variables by simpley setting. to 1? ??(?) =?log. R? ? ??0 ?(?0) · ? ˆ ?(?) ¶ · (B.65) 1 + ·2?2− ?2 1+ 1??2 24 ?? ?2?2(. ) +1 4. 1? (. ) +2 − 3?2 24 ?2 ¸ ?2. + ··· ¾ B.4. Estocástico. model. As originally stated, the SABR model consists of the special case ?(?) = .?ˆ? =?ˆ? ˆ. 1??ˆ. =. ˆ. 2? ˆ?(0) = ? (B.66a) ˆ ?(0) = ? (B.66b) (B.66c) ??1??2=. Making this substitution in. -. shows that the implied normal volatility for this model is ??(?) =??(1 − ?)(? − ?) ?1−?− ?1−? · µ ? ˆ ?(?) ¶ · (B.67a) ½ 1 + ·−?(2 − ?)?2 24?2−2? ? +. 4?1−? ? +2 − 3?2 24 ?2 ¸ ?2. + ··· ¾ through ?(?2), where. =√. as before and (B.67b) ? =? ? ? − ? ? ? ? ˆ ?(?) = log Ãp 33 1 − 2. + ?2−. + ? 1 − ? ! ?
We can simplify this formula by expanding ? −. = p. log. ©1 + 1 24log2. + 1 1920log4. + ··· ? 1 +(1−?)2 24 (B.68a) ?1−?− ?1−?= (1 − ?)(??)(1−?)?2log. and neglecting terms higher than fourth order. This expansion reduces the implied normal volatility to n log2. +(1−?)4 1920log4. + ···(B.68b) ??(?) =. (??)??2 1 + 1 24log2. + log2. +(1−?)4 ·−?(2 − ?)?2 1 1920log4. + ··· 1920log4. + ··· . 4(??)(1−?)?2+2 − 3?2 1 +(1−?)2 24 · µ ? ˆ ?(?) ¸ ¶ · (B.69a) ½ 1 + 24(??)1−?+ 24 ?2 ?2. + ··· ¾ ? where (B.69b) ? =? ?(??)(1−?)?2log. ˆ ?(?) = log Ãp 1 − 2. + ?2−. + ? 1 − ? ! ? This is the formula we use in pricing European calls and puts. To obtain the implied Black volatility, we equate the implied normal volatility. (?) for the SABR model obtained in B.69a - B.69b to the implied normal volatility for Black’s model obtained in B.63. This shows that the implied Black volatility for the SABR model is ??(?) = ?? (??)(1−?)?2 ½ 1 1 +(1−?)2 ·(1 − ?)2?2 24 log2. +(1−?)4 4(??)(1−?)?2+2 − 3?2 1920log4. + ··· · µ ? ˆ ?(?) ¶ ¾ · (B.69c) 1 + 24(??)1−?+ . 24 ?2 ¸ ?2. + ··· ? through ?(?2), where. and ˆ ?(?) are given by B.69b as before. Apart from setting. to 1 to recover the original units, this is the formula quoted in section 2, and fitted to the market in section 3. B.5. Special cases. Two special cases are worthy of special treatment: the stochastic normal model (? = 0) and the stochastic log normal model (? = 1). Both these models are simple enough that the expansion can be continued through ?(?4). For the stochastic normal model (? = 0) the implied volatilities of European calls and puts are ½ µ through ?(?4), where (B.70a) ??(?) =. 1 +2 − 3?2 24 ?2?2. + ··· ¾ (B.70b) ??(?) =. log. ? − ? · ? ˆ ?(?) ¶ · ½ 1 + · ?2 24??+2 − 3?2 24 ?2 ¸ ?2. + ··· ¾ (B.70c) ? =? ? pag. log. ˆ ?(?) = log Ãp 1 − 2. + ?2−. + ? 1 − ? ! ? For the stochastic log normal model (? = 1) the implied volatilities are (B.71a) ??(?) =. − ? log. · µ ? ˆ ?(?) ¶ ·©1 +£−1 24?2+1 4. + 1 24(2 − 3?2)?2¤?2. + ···ª 34
(B.71b) ??(?) =. · µ ? ˆ ?(?) ¶ ·©1 +£1 4. + 1 24(2 − 3?2)?2¤?2. + ···ª through ?(?4), where (B.71c) ? =??log. ˆ ?(?) = log Ãp 1 − 2. + ?2−. + ? 1 − ? ! ? Appendix C. Analysis of the dynamic SABR model. We use effective medium theory [24] to extend the preceding analysis to the dynamic SABR model. As before, we take the volatility ?(?)ˆ. and “volvol” ?(?) to be small, writng ?(?) −→. (?)? and ?(?) −→. (?), and analyze? ˆ? =. (?)ˆ. (ˆ?)??1??ˆ. =. (?)ˆ. 2? (C.1a) (C.1b) with (C.1c) ??1??2= ?(?). in the limit. ¿ 1. We obtain the prices of European options, and from these prices we obtain the implied volatity of these options. After obtaining the results, we replace. (?) −→ ?(?) and. (?) −→ ?(?) to get the answer in terms of the original variables. Suppose the economy is in stateˆ ?(?) =. ˆ ?(?) =. at date. Let. (. ) be the value of, say, a Euro - pean call option with strike. and exercise date. As before, define the transition density ?(. ;. ) by n and define Z∞ Repeating the analysis in Appendix B through equation B.10a, B.10b now shows that the option price is given by (C.2a) ?(. ;. ). ≡ prob . ˆ?(?). +. ˆ ?(?). +. ¯¯¯ˆ?(?) =. ˆ ?(?) = ? o (C.2b) ?(. ;. ) = −∞ ?2?(. ;. ). (C.3) ? (. ) = [? − ?]++1 2?2?2(?) Z. ? ?2(?)?(. ;. ). where ?(. ;. ) is the solution of the backwards problem ??+1 2?2©?2?2?2(?). + 2. 2?(?). + ?2?2. ª= 0? for. (C.4a) ? = ?2?(? − ?)? for. =. (C.4b) We eliminate ?(?) by defining the new time variable (C.5) ? = Z? 0 ?2(?0)??0??0= Z? 0 ?2(?0)??0. = Z. 0 ?2(?0)??0? 35
Then the option price becomes (C.6) ? (. ) = [? − ?]++1 2?2?2(?) Z. ? ?(. ;?0??)??0? where ?(. ;?0??) solves the forward problem 2?2©?2?2(?). + 2?(?)?2?(?). + ?2(?)?2. (C.7b) ??+1 ª= 0for. 0 for. = ?0? (C.7a) ? = ?2?(? − ?)? Here (C.8) ?(?) = ?(?)?(?)??(?)??(?) = ?(?)??(?)? We solve this problem by using an effective media strategy [24]. In this strategy our objective is to determine which constant values ¯. and ¯. yield the same option price as the the time dependent coefficients ?(?) and ?(?). If we could find these constant values, this would reduce the problem to the non-dynamic SABR model solved in Appendix B. We carry out this strategy by applying the same series of time-independent transformations that was used to solve the non-dynamic SABR model in Appendix B, defining the transformations in terms of the (as yet unknown) constants ¯. and ¯. The resulting problem is relatively complex, more complex than the canonical problem obtained in Appedix B. We use a regular perturbation expansion to solve this problem, and once we have solved this problem, we choose ¯. and ¯. so that all terms arising from the time dependence of ?(?) and ?(?) cancel out. As we shall see, this simultaneously determines the “effective” parameters and allows us to use the analysis in Appendix B to obtain the implied volatility of the option. C.1. Transformation. As in Appendix B, we change independent variables to (C.9a) ? = 1 ?? Z? ? ??0 ?(?0)? and define (C.9b) ?(. ) = ?(?)? We then change dependent variables from. toˆ. and then to. (C.9c) ˆ? =? ??(?)?? (C.9d) ? = p?(?)??(?)ˆ? ≡ p?(0)??(. )ˆ?? Following the reasoning in Appendix B, we obtain (C.10) ? (. ) = [? − ?]++1 2??p?(0)?(. ) Z. ? ?(. ;?0)??0? where ?(. ;?0) is the solution of (C.11a) ??+1 2 ¡1 − 2. + ?2?2?2¢. −1 2?2. 0 ?(. − ?) + ?2?2 µ 1 4 ?00 ? −3 8 ?02 ?2 ¶ ? = 0 36
for. 0, and (C.11b) ? = ?(?) at. = ?0 through ?(?2). See B.29, B.31a, and B.31b. There are no. derivatives in equations C.11a, C.11b, so we can treat. as a parameter instead of a variable. Through ?(?2) we can also treat ?0. and ?00. as constants: (C.12) ?1≡?0(. 0) ?(. 0)??2≡?00(. 0) ?(. 0)? where ?0will be chosen later. Thus we must solve ¡1 − 2. + ?2?2?2¢. −1 (C.13b) (C.13a) ??+1 22?2. 1(. − ?) + ?2?2¡1 ? = ?(?) at. = ?0? 4?2−3 8?2 1 ¢? = 0for. 0? At this point we would like to use a time-independent transformation to remove the. term from equation C.13a. It is not possible to cancel this term exactly, since the coefficient ?(?) is time dependent. Instead we use the transformation (C.14) ? = ? 1 4?2??1??2ˆ. where the constant. will be chosen later. This transformation yields ˆ. +1 2 ¡1 − 2. + ?2?2?2¢ˆ. −1 +1 2?2??1(? − ?)?ˆ. 4?2??1(2? + ?)ˆ. + ?2?2¡1 ˆ. = ?(?) at. = ?0? 4?2−3 8?2 1 ¢ˆ. = 0for. 0? (C.15a) through ?(?2). Later the constant. will be selected so that the change in the option price caused by the term transformation cancels out the. term “on average.” In a similar vein we define p and Z?¯. where the constants ¯. and ¯. will be chosen later. This yields 1 2?2??1??ˆ. is exactly offset by the change in price due to 1 2?2??1??ˆ. término. In this way to the (C.16a) ?(?¯. ) = 1 − 2?¯. + ?2¯ ?2?2? (C.16b) ? = 1 ?¯ ? 0 ?? ?(?)= 1 ?¯?log( p1 − 2?¯. + ?2¯ ?2?2− ¯. ¯. + ?¯. 1 − ¯. ¯ ? )? (C.17a) ˆ. +1 2 1 − 2. + ?2?2?2 1 − 2?¯. + ?2¯ ?2?2(ˆ. − ?¯. 0(?¯. )ˆ. ) −1 2?2??1(? − ?)?ˆ. +1 4?2??1(2? + ?)ˆ. + ?2?2¡1 4?2−3 8?2 1 ¢ˆ. = 0for. 0? 37
(C.58d) ¯? = R? 0˜ ?[?2(˜ ?) − ¯ ?2]?˜ ? 1 2?2 ? Equivalently, the option prices are given by Black’s formula with the effective Black volatility of µ ½ ??(?) =?log. R? 1 + ? ??0 ?(?0) ·2?2− ?2 · ? ˆ ?(?) ¶ · (C.59) 1+ 1??2 24 ?? ?2?2(. ) +1 4¯. 1? (. ) +2¯ ?2− 3¯ ?2 24 +1 2¯? ¸ ?2. + ··· Appendix D. Analysis of other stochastic vol models. Adapt analysis to other SV models. Just quote results? Appendix E. Analysis of other stochastic vol models. Adapt analysis to other SV models. Just quote results? REFERENCES [1] D. T. Breeden and R. H. Litzenberger, Prices of state-contingent claims implicit in option prices, J. Business, 51 (1994), pp. 621-651. [2] B. Dupire, Pricing with a smile, Risk, Jan. 1994, pp. 18—20. [3] B. Dupire, Pricing and hedging with smiles, in Mathematics of Derivative Securities, M. A. H. Dempster and S. R. Pliska, eds. Cambridge University Press, Cambridge, 1997, pp. 103—111 [4] E. DermaN and I. Kani, Riding on a smile, Risk, Feb. 1994, pp. 32—39. [5] E. Derman and I. Kani, Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility, Int J. Theor Appl Finance, 1 (1998), pp. 61—110. [6] J. M. Harrison and S. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. Appl, 11 (1981), pp. 215-260. [7] J. M. Harrison and D. Krebs, Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory, 20 (1979), pp. 381-408. [8] I. Karatzas, J. P. Lehoczky, S. E. Shreve, and G. L. Xus, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim, 29 (1991), pp. 702—730. [9] J. Michael Steele, Stochastic Calculus and Financial Applications, Springer, 2001 [10] F. Jamshidean, Libor and swap market models and measures, Fin. Stoch. 1 (1997), pp. 293-330 [11] F. Black, The pricing of commodity contracts, Jour. Pol. Ec. 81 (1976), pp. 167-179. [12] John C. Hull, Options, Futures, and Other Derivative Securities, Prentice Hall, 1997. [13] P. Wilmott, Paul Wilmott on Quantitative Finance, John Wiley & Sons, 2000. [14] Patrick S. Hagan and Diana E. Woodward, Equivalent Black volatilities, App. Math. Finance, 6 (1999), pp. 147—157. [15] P. S. Hagan, A. Lesniewski and D. E. Woodward, Geometric optics in finance, in preparation. [16] F. Wan, Mathematical Models and Their Analysis, Harper-Row, 1989. [17] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. of Finance, 42 (1987), pp. 281-300. [18] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev of Fin Studies, 6 (1993), pp. 327-343. [19] A. Lewis, Option Valuation Under Stochastic Volatility, Financial Press, 2000. [20] J. P. Fouque, G. Papanicolaou, K. R. Sirclair, Derivatives in Financial Markets with Stochastic Volatility, Cambridge Univ Press, 2000. [21] N. A. Berner, Hedging vanna & volga, DKW, private communicatons [22] J. D. Cole, Perturbation Methods in Applied Mathematics, Ginn-Blaisdell, 1968. [23] J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, 1985. [24] J. F. Clouets, Diffusion Approximation of a Transport Process in Random Media, SIAM J Appl Math, 58 (1998), pp. 1604—1621. [25] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1988. [26] B. Okdendal, Stochastic Differential Equations, Springer, 1998. [27] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 1998. [28] G. B. Whitham, Linear and Nonlinear Waves, Wiley, 1974. [29] J. C. Neu, Thesis, California Institute of Technology, 1978 44
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Stochastic Alpha-Beta-Rho Hedging for Foreign Exchange Options: Is It Worth the Effort?
Yifan Yang, Frank J. Fabozzi, and Michele Leonardo Bianchi
To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals. com or 212-224-3675.
This article applies the stochastic alpha-beta-rho (SABR) model to the foreign exchange options market. The model pricing and hedging performance are tested using three years of historical data from August 2, 2010 to July 31, 2017 for the four most traded currency pairs. The results are compared to the hedging performance of the Black–Scholes model. The empirical study shows that the SABR model can fit and predict market volatility well. However, the hedging results show that the SABR model does not provide a more accurate hedge ratio than the Black–Scholes model. This finding is surprising given the well-known criticism of the Black–Scholes model.
Yifan Yang is a quantitative analyst at Noble Group in Singapore. yifan. brian@gmail. com
Frank J. Fabozzi is a professor of finance at EDHEC Business School in Nice, France. frank. fabozzi@edhec. edu
Michele Leonardo Bianchi
Michele Leonardo Bianchi is a researcher in the Macroprudential Analysis Division of the Regulation and Macroprudential Analysis Directorate at the Bank of Italy in Rome, Italy. micheleleonardo. bianchi@bancaditalia. it
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Paulot. L. “Asymptotic Implied Volatility at the Second Order with Application to the SABR Model.” Working paper, Sophis Technology, 2009, http://arxiv. org/pdf/0906.0658.pdf.
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Article Tools
Consistent Pricing of FX Options
Antonio Castagna
Fabio Mercurio
In the current markets, options with different strikes or maturities are usually priced with different implied volatilities. This stylized fact, which is commonly referred to asfsmile effect, can be accommodated by resorting to specific models, either for pricing exotic derivatives or for inferring implied volatilities for non quoted strikes or maturities. The former task is typically achieved by introducing alternative dynamics for the underlying asset price, whereas the latter is often tackled by means of statical adjustments or interpolations.
In this article, we deal with this latter issue and analyze a possible solution in a foreign exchange (FX) option market. In such a market, in fact, there are only three active quotes for each market maturity (the 0Delta straddle, the risk reversal and the vega-weighted butterfly), thus presenting us with the problem of a consistent determination of the other implied volatilities.
FX brokers and market makers typically address this issue by using an empirical procedure to construct the whole smile for a given maturity. Volatility quotes are then provided in terms of the option's Delta, for ranges from the 5Delta put to the 5Delta call.
In the following, we will review this market procedure for a given currency. In particular, we will derive closed-form formulas so as to render its construction more explicit. We will then test the robustness (in a static sense) of the resulting smile, in that changing consistently the three initial pairs of strike and volatility produces eventually the same implied volatility curve. We will also show that the same procedure applied to Europeanstyle claims is consistent with static-replication results and consider, as an example, the practical case of a quanto European option. We will finally prove that the market procedure can also be justified in dynamical terms, by defining a hedging strategy that is locally replicating and self-financing.
Number of Pages in PDF File: 15
Keywords: FX option, smile, consisten pricing, stochastic volatility
JEL Classification: G13
Date posted: January 5, 2006
Repricing the Cross Smile: An Analytic Joint Density
Peter Austing
Derivative contracts on multiple foreign exchange rates must be priced to avoid arbitrage by contracts on the cross-rates. Given the triangle of smiles for two underlyings and their cross, we provide an analytic formula for a joint probability density such that all three vanilla markets are repriced. The method extends to N dimensions and leads to simple necessary conditions for a triangle of smiles to be arbitrage-free in the model.
Number of Pages in PDF File: 13
Keywords: Multi-currency model. FX volatility smile, Cross smile, Best-of option, Worst-of option, Basket option
Date posted: July 10, 2011
Cita Sugerida
Austing, Peter, Repricing the Cross Smile: An Analytic Joint Density (May 17, 2011). Risk, July 2011. Available at SSRN: http://ssrn. com/abstract=1882343
FX Smile in the Heston Model
Albrecher, H. Mayer, P. Schoutens, W. and Tistaert, J. (2006). The little Heston trap, Wilmott Magazine, January: 83–92. 231–262.
Andersen, L. and Andreasen, J. (2000). Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing, Review of Derivatives Research 4: 231–262.
Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model, The Journal of Computational Finance 11(3), 1–42.
Apel, T. Winkler, G. and Wystup, U. (2002). Valuation of options in Heston’s stochastic volatility model using finite element methods, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk, Risk Books, London.
Bakshi, G. Cao, C. and Chen, Z. (1997). Empirical Performance of Alternative Option Pricing Models, Journal of Finance 52: 2003–2049.
Bates, D. (1996). Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, Review of Financial Studies 9: 69–107.
Borak, S. Detlefsen, K. and Hardle, W. (2005). FFT-based option pricing, in P. Cizek, W. Hardle, R. Weron (eds.) Statistical Tools for Finance and Insurance, Springer, Berlin.
Broto, C. and Ruiz, E. (2004). Estimation methods for stochastic volatility models: A survey, Journal of Economic Surveys 18(5): 613–649.
Carr, P. and Madan, D. (1999). Option valuation using the fast Fourier transform, Journal of Computational Finance 2: 61–73.
Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes, Chapman & Hall/CRC.
Derman, E. and Kani, I. (1994). Riding on a Smile, RISK 7(2): 32–39.
Dragulescu, A. A. and Yakovenko, V. M. (2002). Probability distribution of returns in the Heston model with stochastic volatility, Quantitative Finance 2: 443–453.
Dupire, B. (1994). Pricing with a Smile, RISK 7(1): 18–20.
Eberlein, E. Kallsen, J. and Kristen, J. (2003). Risk Management Based on Stochastic Volatility, Journal of Risk 5(2): 19–44.
Fengler, M. (2005). Semiparametric Modelling of Implied Volatility, Springer.
Fouque, J.-P. Papanicolaou, G. and Sircar, K. R. (2000). Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press.
Garman, M. B. and Kohlhagen, S. W. (1983). Foreign currency option values, Journal of International Monet & Finance 2: 231–237.
Gatheral, J. (2006). The Volatility Surface: A ractitioner’s Guide, Wiley.
Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, Springer-Verlag, New-York.
Hakala, J. and Wystup, U. (2002). Heston’s Stochastic Volatility Model Applied to Foreign Exchange Options, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk, Risk Books, London.
Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6: 327–343.
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Jeanblanc, M. Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets, Springer.
Kluge, T. (2002). Pricing derivatives in stochastic volatility models using the finite difference method, Diploma thesis, Chemnitz Technical University.
Lee, R. (2004). Option pricing by transform methods: extensions, unification and error control, Journal of Computational Finance 7(3): 51–86.
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International Journal of Dental and Health Sciences
Welcome to International Journal of Dental and Health Sciences [IJDHS]
About IJDHS
The International Journal of Dental and Health Sciences is an international peer-reviewed bi-monthly indexed dental and medical journal. The mission of International Journal of Dental and Health Sciences is to provide its readers with up-to-date information relevant to Dental and Health sciences. The journal policy is to publish work deemed by peer reviewers to be a coherent and sound addition to scientific knowledge and to put less emphasis on interest levels, provided that the research constitutes a useful contribution to the field. The Journal publishes original research papers, review articles, case studies, reports and technical note in all aspects of Dental and Health sciences.
Our journal is open access, which means that your paper is available to anyone in the world to download for free directly from our website.
Indexed in various international database and university libraries.
Fast and efficient on-line submission.
We have one of the fastest turn around times of any publisher in the world. Generally peer review is complete within 2 weeks and the editor’s decision within 24 hours of this. It is therefore very rare to have to wait more than 3 weeks for a final decision.
Although peer review is rapid it is also very thorough. Many authors have found that our peer reviewer’s comments substantially add to their final papers.
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The amount of article processing charges (APC) will be established in the yearly classification of the World Bank Economies of countries (i. e. Low income countries, Middle income countries and High Income countries).
For International Journal of Dental and Health Sciences, authors are asked to pay a fee of $50 or $100 per processed paper according to classification of the World Bank Economies of countries (i. e. Low /Middle income countries and High Income countries respectively), but only if the article is accepted for publication in this journal after peer-review and possible revision of the manuscript. Note that many national and private research funding organizations and universities explicitly cover such fees for articles originated in funded research projects.
The Front Office Manual
The Definitive Guide to Trading, Structuring and Sales
Andrew Sutherland and Jason Court
Can you describe the rules for settling foreign exchange in Jordan? What is strange about the Australian swap market? What are IMM dates, and why do credit derivatives abuse them? Financial markets are exceedingly complex, often to their own detriment. The building blocks of finance are quite simple, however. Complexity arises when these building blocks are traded, combined, and modeled using a myriad of conventions, and using market knowledge which is obscure and hard for outsiders to discover. The Front Office Manual is a practical introduction to the front office, guiding readers through the functions and financial instruments commonly encountered in an investment banking business and importantly, how they work and are implemented in practice. The book begins with a guided tour of the front office, and of how a trade, the life-blood of all investment banks, actually works from inception, through its mid life-cycle events through to termination. The book also introduces the various market participants – investors, hedge funds, banks and asset managers to name but a few – with whom the front office interacts. Finally, the book describes the wide range of financial products used in trading and structuring, providing detail on the products themselves and the techniques needed to produce prices, estimate risk, and produce meaningful analysis. Throughout the book, the focus is on practical implementation, and 'how things are actually done in a banking environment', making this an invaluable working manual for newcomers to the industry, and those looking to move from a middle or back office function, or a different part of the financial community, to the investment banking front office.
How to cite this book (export citation )
Harvard Andrew Sutherland and Jason Court. (November 2017). The Front Office Manual . [Online] Available at: http://www. palgraveconnect. com/pc/doifinder/10.1057/9781137030696.0001. (Accessed: 17 March 2017). APA Andrew Sutherland and Jason Court. (November 2017). The Front Office Manual . Retrieved from http://www. palgraveconnect. com/pc/doifinder/10.1057/9781137030696.0001 MLA Andrew Sutherland and Jason Court. The Front Office Manual . (November 2017) Palgrave Macmillan. 17 March 2017. Vancouver Andrew Sutherland and Jason Court. The Front Office Manual [internet]. Basingstoke: Palgrave Macmillan; November 2017. [cited 2017 March 17]. Available from: http://www. palgraveconnect. com/pc/doifinder/10.1057/9781137030696.0001 OSCOLA Andrew Sutherland and Jason Court. The Front Office Manual . Palgrave Macmillan November 2017
Chapter 4. The Mechanics of Simple Yield Curve Construction
Deventer, K. J. (1994) ‘Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness’, The Journal of Fixed Income, 52–62. » Search Google
Patrick S. Hagan, G. W. (2008) Methods for Constructing a Yield Curve, WILMOTT magazine. » Search Google
Ron, U. (2000) ‘A Practical Guide to Swap Curve Construction’, Bank of Canada Working Paper 2000–17. » Search Google
West, G. (n. d.) ‘A brief comparison of interpolation methods for yield curve construction’, http://www. finmod. co. za. » Search Google
Chapter 5. Discount and Forward Interest Rate Curves
Deventer, K. J. (1994) ‘Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness’, The Journal of Fixed Income, 52–62. » Search Google
Patrick S. Hagan, G. W. (2008) ‘Methods for Constructing a Yield Curve’, WILMOTT Magazine. » Search Google
Ron, U. (2000) ‘A Practical Guide to Swap Curve Construction’, Bank of Canada Working Paper 2000–17. » Search Google
West, G. (n. d.) ‘A brief comparison of interpolation methods for yield curve construction’, http://www. finmod. co. za. » Search Google
Chapter 7. Cross-Currency Trades in the Future: Forward FX and Cross-Currency Basis Swaps
Masaaki Fujii, Y. S. (2010) ‘A Note on Construction of Multiple Swap Curves with and without Collateral’, Financial Services Agency, Government of Japan. » Search Google
Masaaki Fujii, Y. S. (2010) ‘On the Term Structure of Interest Rates with Basis Spreads, Collateral and Multiple Currencies’, Bank of Japan. » Search Google
Chapter 8. Basic Equity Trades
Bank of England (2010) ‘Securities Lending: An Introductory Guide’, http://www. bankofengland. co. uk/markets/Documents/gilts/sl_intro_green_9_10.pdf. » Search Google
European Central Bank (2011) ‘Settlement Fails – Report on Securities Settlement Systems (SSS) Measures to Ensure Timely Settlement’, http://www. ecb. int/pub/pdf/other/settlementfails042011en. pdf. » Search Google
Faulkner, M. C. (2007) ‘An Introduction to Securities Lending’, http://www. bankofengland. co. uk/markets/Documents/gilts/securitieslending. pdf. » Search Google
International Securities Lending Association (2010) Global Master Securities Lending Agreement. » Search Google
Chapter 9. Government Bonds
Choudhry, M. (2002) The Repo Handbook, Butterworth-Heinemann. » Search Google Books » Buscar en una biblioteca
Duffie, D. (1996) ‘Special Repo Rates’, Journal of Finance, 493–525. » Search Google
Place, J. (2000) Basic Bond Analysis (Handbooks in Central Banking no. 20), London: Bank of England. » Search Google Books » Buscar en una biblioteca
Chapter 11. Vanilla Options
Avellaneda, M. (n. d.) ‘Trinomial Trees and Finite-Difference Schemes’, http://www. math. nyu. edu/faculty/avellane/. » Enlazar
Homescu, C. (n. d.) Implied Volatility Surface: Construction Methodologies and Characteristics. » Search Google
Wystup, U. (n. d.) ‘FX Smile Modelling’, Wiley Encyclopedia of Quantitative Finance. » Search Google
Chapter 12. More Vanilla Options
Antonio Castagna, F. M. (n. d.) ‘Consistent Pricing of FX Options’, http://www. fabiomercurio. it. » Enlazar
Castagna, A. (2009) FX Options and Smile Risk, Wiley. » Search Google Books » Buscar en una biblioteca
Dimitri Reiswich, U. W. (n. d.) ‘FX Volatility Smile Construction’, No 20, CPQF Working Paper Series from Frankfurt School of Finance and Management, Centre for Practical Quantitative Finance (CPQF). » Search Google Books » Buscar en una biblioteca
Chapter 13. Vanilla Interest Rate Options
Bartlett, B. (n. d.) ‘Hedging under SABR Model’, Wilmott Magazine. » Search Google
Fabio Mercurio, D. B. (2006) Interest Rate Models – Theory and Practice, Springer Finance. » Search Google Books » Buscar en una biblioteca
Graeme West, L. W. (n. d.) ‘Introduction to Black’s Model for Interest Rate Derivatives’, http://www. finmod. co. za. » Enlazar
Hagan, P. S. (March/April 2003) ‘Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors’, Wilmott Magazine. » Search Google
Henrard, M. (2011) ‘Swaptions: 1 Price, 10 Deltas, and. 6 1/2 Gammas’, Wilmott Magazine. » Search Google
Henrard, M. (n. d.) ‘Cash Settled Swaptions: How Wrong are We?’, Available at SSRN: http://ssrn. com/abstract=1703846. » Enlazar
Obloj, J. (n. d.) ‘Fine Tune your Smile’, arXiv:0708.0998 (http://arxiv. org/abs/0708.0998v3). » Enlazar
Patrick Hagan, D. K. (2002) ‘Managing Smile Risk’, Wilmott Magazine. » Search Google
Infomation about the author(s)
Andrew Sutherland is a finance enthusiast with seventeen years of experience in the industry. After obtaining his law degree from the University of Virginia in 1995, Andrew became a derivatives technologist for Citibank in New York. From New York, he moved to London, where he became a structured products trader for Citi, working with interest rate products and FX. He has since become a front-office technology expert, leading teams to develop complex front-office systems for a variety of institutions, including Barclays and HSBC. He lives in London with his wife and two children.
Jason Court is a director of boutique consultancy Jmoni Limited. He started his career in the City of London in 1986 working in operations for Midland Bank International (now HSBC) before moving into corporate treasury sales, specialising in rates and foreign exchange. In the mid-1990s he moved into information technology as an analyst, leveraging his detailed front to back process and product knowledge. He has worked at Baring Brothers (now ING Barings) in debt derivatives, Merrill Lynch (now Bank America Merrill Lynch) where he was Head of Emerging Markets Technology, and Credit Suisse where he ran equity derivatives technology in EMEA. He is an avid cricket fan, and when not working enjoys spending time at the Oval watching Surrey CCC with his three children.
Prelims
Volatility Smile Explained
In finance, the volatility smile is a long-observed pattern in which at-the-money options tend to have lower implied volatilities than in - or out-of-the-money options. The pattern displays different characteristics for different markets and results from the probability of extreme moves
The Above video explains a plot of implied volatility (i. e. the volatility that forces the BSM model option price to equal the observed market price) against strike price. The smile is proof the model is imprecise (incorrect in some assumption); p. ej. returns are not lognormally distributed
When implied volatility is plotted against strike price, the resulting graph is typically downward sloping for equity markets, or valley-shaped for currency markets. For markets where the graph is downward sloping, such as for equity options, the term “volatility skew” is often used. For other markets, such as FX options or equity index options, where the typical graph turns up at either end, the more familiar term “volatility smile” is used. For example, the implied volatility for upside (i. e. high strike) equity options is typically lower than for at-the-money equity options. However, the implied volatilities of options on foreign exchange contracts tend to rise in both the downside and upside directions. In equity markets, a small tilted smile is often observed near the money as a kink in the general downward sloping implicit volatility graph. Sometimes the term “smirk” is used to describe a skewed smile.
Chart shown above is the Volatility smile plot for Suzlon Energy Limited for both the call and put option for Aug 2010 options series
About Rajandran
Rajandran is a trading strategy designer and founder of Marketcalls, a hugely popular trading site since 2007 and one of the most intelligent blog in the world to share knowledge on Technical Analysis, Trading systems & Estrategias de negociación.
Comentarios
Required US Government Disclaimer & CTFC Rule 4.41
Futures trading contains substantial risk and is not suitable for every investor. Un inversionista podría perder todo o más de la inversión inicial. Capital de riesgo es el dinero que se puede perder sin poner en peligro la seguridad financiera o el estilo de vida. Sólo considerar el capital de riesgo que debe ser utilizado para el comercio y sólo aquellos con suficiente capital de riesgo debe considerar la negociación. El rendimiento pasado no es necesariamente indicativa de resultados futuros. CTFC RULE 4.41 – HYPOTHETICAL OR SIMULATED PERFORMANCE RESULTS HAVE CERTAIN LIMITATIONS. DESCONOCIDO UN REGISTRO DE RENDIMIENTO REAL, LOS RESULTADOS SIMULADOS NO REPRESENTAN COMERCIO REAL. TAMBIÉN, DADO QUE LOS COMERCIOS NO HAN SIDO EJECUTADOS, LOS RESULTADOS PUEDEN TENERSE COMPARTIDOS POR EL IMPACTO, SI CUALQUIERA, DE CIERTOS FACTORES DE MERCADO TALES COMO LA LIQUIDEZ. LOS PROGRAMAS DE COMERCIO SIMULADOS EN GENERAL ESTÁN SUJETOS AL FACTOR DE QUE SEAN DISEÑADOS CON EL BENEFICIO DE HINDSIGHT. NO SE HACE NINGUNA REPRESENTACIÓN QUE CUALQUIER CUENTA TENDRÁ O ES POSIBLE PARA LOGRAR GANANCIAS O PÉRDIDAS SIMILARES A LOS MOSTRADOS. All trades, patterns, charts, systems, etc. discussed in this website or advertisement are for illustrative purposes only and not construed as specific advisory recommendations. Todas las ideas y materiales presentados aquí son para propósitos informativos y educativos solamente. Nunca se ha desarrollado ningún sistema o metodología comercial que pueda garantizar beneficios o evitar pérdidas. The testimonials and examples used herein are exceptional results which do not apply to average people and are not intended to represent or guarantee that anyone will achieve the same or similar results. Las operaciones que se basan en la dependencia de los sistemas de Trend Methods se toman bajo su propio riesgo para su propia cuenta. No se trata de una oferta de compra o venta de futuros.
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FX Option Performance: Long Term Anomalies, Opportunities and Hedging Strategies by Jessica James
Complimentary Book:
Complimentary book included with this course: FX Option Performance: An Analysis of the Value Delivered by FX Options since the Start of the Market
Outline: Why take this course?
Another course on FX options? But surely there are plenty of courses and books which tell you how to price and hedge these products? Indeed there are – but that’s not what this is for. On this course you will learn about the value, not the price, of FX options.
The FX Option market has grown from humble beginnings in the 1980s to a global behemoth, with hundreds of billions of US dollars worth of flow going through it each trading day. With such size and liquidity, it is generally assumed to be efficient, with few profit opportunities. But astonishingly, this is not so. Jessica James, the course leader, presents new work where she has taken a completely different attitude to most option studies; instead of looking into the theory of how the products are valued and what they should be worth, she has gone back through the historical data to find out what they actually delivered compared to what they cost.
This course thus offers a unique and practical guide to option trading and hedging, having the courage to report how much these contracts have really made or lost. This critical (and often overlooked) information gives investors and hedgers alike the edge they need to make more informed decisions. Presented in accessible terms and not overly technical, with over a hundred charts and diagrams to illustrate the findings, this course should appeal to anyone with an interest in the area. Jessica reveals where to look for value and offers help to corporations hedging their FX exposures. Some of the results are truly remarkable; for example, longer dated call options pay back on average only about half of their initial cost. It seems astonishing that this information is not public knowledge.
The course will be a fascinating introduction for students entering the field, a source of exciting ideas for investors looking for opportunities, and an essential guide for corporations wondering how to hedge their FX exposures.
Day 1: ‘The FX Option Market: How it works, what’s right, and what’s wrong’
(1) Introduction to the FX option market
History of theory and trading
Market Participants
What they are, why buy or sell them
How to price them
Black-Scholes-Merton
Limitations of this classical model
The underlyings (FX rates, depos, forwards, vols) and how they behave. (correlation of underlyings, implied vs realised quantities)
How they affect the price of options – lots of market examples, EURCHF will be fun
The ‘greeks’
How options are traded
By a hedger
By a hedge fund
By a trading desk
(2) What’s wrong with the market: Puts vs Calls
What’s the ‘fair price’ for an option?
Backtesting FX option returns – theory, data, difficulties
How might mispricings arise?
Broad brush results: straddles
Puts vs calls: first sign that something is wrong
Individual ccy results – clues to what is happening with AUD, JPY
Explanation and theory of the carry problem
Implications for hedgers
Trading strategies
Day 2: ‘The FX Option Market: Long term anomalies and how to use them for trading and hedging’
(1) What’s wrong with the market: ATMF vs OTM options, G10 vs EM
Theory of OTM options – volatility smile
Data available (risk reversals and butterflies) and what does it mean
Illustration that OTM options are poor value
Detailed discussion as to how/why
Differences with G10 and EM
ATMF
OTM
With tenor
Theory of differences
Currency-by-currency discussion
Trading strategies
(2) The FX Carry Trade
Historia
Theory
Improvements/enhancements. How to reduce risk and improve returns
Can we do it with options?
Trading strategies
Fee Structure
This workshop is free online via The Quants Hub Annual Subscription Service!
Standard Fee to attend the Live event: £599 + UK VAT per day (Register to BOTH days of the workshop and receive £200 discount) Register Below to the Live event
All attendees either online or at the live event in London will receive the recorded video workshop!
About the Presenter
Jessica James is Head of the FX Quantitative Solutions team at Commerzbank in London. She joined Commerzbank from Citigroup where she held a number of FX roles, latterly as Global Head of the Quantitative Investor Solutions Group. Prior to this she was the Head of Risk Advisory and Currency Overlay Team for Bank One. Before her career in finance, James lectured in physics at Trinity College, Oxford.
Her significant publications include the ‘Handbook of Foreign Exchange’ (Wiley), 'Interest Rate Modelling' (Wiley), and 'Currency Management' (Risk books). Her new book ‘FX Option Performance’ will be published April 2017. She has been closely associated with the development of currency as an asset class, being one of the first to create overlay and currency alpha products.
Jessica is on the Board of the Journal of Quantitative Finance, and is a Visiting Professor both at UCL and at Cass Business School. She is a Managing Editor for the Journal of Quantitative Finance. Apart from her financial appointments, she is a Fellow of the Institute of Physics and has been a member of their governing body and of their Industry and Business Board.
Detalles
Ubicación:
Radisson Blu Portman Hotel 22 Portman Square London W1H 7BG Hotel Website
Flight details:
All delegates flying into London on the morning of the event are reminded that they should arrive 30 minutes before the workshop starts for registration. The hotels West End location is approximately 1 hour from all 3 main London airports, Heathrow, Gatwick and City. Returning flights should equally allow for the events finishing time.
Sponsorship:
World Business Strategies offer sponsorship opportunities for all events, E-mail headers and the web site. Contact Sponsorship: +44 (0) 1273 201752
Renuncia:
World Business Strategies command the rights to cancel or alter any part of this programme.
Cancellation:
By completing of this form the client hereby enters into a agreement stating that if a cancellation is made by fax or writing within two weeks of the event date no refund shall be given. However in certain circumstances a credit note maybe issued for future events.
Prior to the two week deadline, cancellations are subject to a fee of 25% of the overall course cost.
Quants Hub Subscription Service
Montly Subscription Service
1 Month Service: £899 (Workshops only, no programming school courses) 3 Months Service: £1499 (Workshop access all areas & 1 Programming School course) 6 Months Service: £1999 (Workshop access all areas & 2 Programming School courses)
Programming School Only Annual Subscription Service
This subscription option allows you to access 4 programming school start date courses per annum, plus all self-paced courses: £2499
Workshop Only Annual Subscription Service
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Heston's Model and the Smile
Bibliografía
Andersen, L. and Andreasen, J. (2000). Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing, Review of Derivatives Research 4 . 231–262. CrossRef
Andreasen, J. (1997). A Gaussian Exchange Rate and Term Structure Model, Essays on Contingent Claim Pricing 97/2 . PhD thesis.
Apel, T. Winkler, G. and Wystup, U. (2002). Valuation of options in Heston's stochastic volatility model using finite element methods, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk . Risk Books, London.
Bakshi, G. Cao, C. and Chen, Z. (1997). Empirical Performance of Alternative Option Pricing Models, Journal of Finance 52 . 2003–2049.
Barndorff-Nielsen, O. E. Mikosch, T. and Resnick, S. (2001). Levy processes: Theory and Applications . Birkhäuser.
Bates, D. (1996). Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, Review of Financial Studies 9 . 69–107. CrossRef
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Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes . Chapman & Hall/CRC.
Cox, J. C. Ingersoll, J. E. and Ross, S. A. (1985). A Theory of the Term Structure of Interest Rates, Econometrica 53 . 385–407. MathSciNet
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Eberlein, E. Kallsen, J. and Kristen, J. (2003). Risk Management Based on Stochastic Volatility, Journal of Risk 5(2) . 19–44.
Fengler, M. (2005). Semiparametric Modelling of Implied Volatility . Springer.
Garman, M. B. and Kohlhagen, S. W. (1983). Foreign currency option values, Journal of International Monet & Finance 2 . 231–237. CrossRef
Hakala, J. and Wystup, U. (2002). Heston's Stochastic Volatility Model Applied to Foreign Exchange Options, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk . Risk Books, London.
Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6 . 327–343. CrossRef
Hull, J. and White, A. (1987). The Pricing of Options with Stochastic Volatilities, Journal of Finance 42 . 281–300.
Kluge, T. (2002). Pricing derivatives in stochastic volatility models using the finite difference method . Diploma thesis, Chemnitz Technical University.
Martinez, M. and Senge, T. (2002). A Jump-Diffusion Model Applied to Foreign Exchange Markets, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk . Risk Books, London.
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Tompkins, R. G. and D'Ecclesia, R. L. (2004). Unconditional Return Disturbances: A Non-Parametric Simulation Approach, Journal of Banking and Finance . to appear.
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Una relación de deuda y rentabilidad utilizada para determinar la facilidad con que una empresa puede pagar intereses sobre la deuda pendiente.
Una cuenta que se puede encontrar en la parte de activos del balance de una empresa. La buena voluntad a menudo puede surgir cuando una empresa.
Un fondo de índice es un tipo de fondo mutuo con una cartera construida para igualar o rastrear los componentes de un índice de mercado, tales.
Un contrato de derivados mediante el cual dos partes intercambian instrumentos financieros. These instruments can be almost anything.
Aprenda lo que es EBITDA, vea un video corto para aprender más y con lecturas le enseñamos cómo calcularlo usando MS.
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Arbitrage-Free Prediction of the Implied Volatility Smile
Petros Dellaportas
Athens University of Economics and Business
Aleksandar Mijatovic
Imperial College London
This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to n strikes (with n large, e. g. n ≥ 40) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any n-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n ∈ N in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is not arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts, suggesting a potential for building management strategies for portfolios of derivative products, akin to the ones widely used in the underlying equity and futures markets.
Number of Pages in PDF File: 18
Keywords: Prediction of option prices in FX, Risk-neutral measure, Implied Volatility, trading strategy for options
Date posted: July 23, 2017
Cita Sugerida
Dellaportas, Petros and Mijatovic, Aleksandar, Arbitrage-Free Prediction of the Implied Volatility Smile (March 22, 2017). Risk Magazine, Forthcoming. Available at SSRN: http://ssrn. com/abstract=2469770 or http://dx. doi. org/10.2139/ssrn.2469770
FX volatility smile construction
Dimitri Reiswich and Uwe Wystup
Abstract: The foreign exchange options market is one of the largest and most liquid OTC derivative markets in the world. Surprisingly, very little is known in the academic literature about the construction of the most important object in this market: The implied volatility smile. The smile construction procedure and the volatility quoting mechanisms are FX specific and differ significantly from other markets. We give a detailed overview of these quoting mechanisms and introduce the resulting smile construction problem. Furthermore, we provide a new formula which can be used for an efficient and robust FX smile construction.
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Home | Instructors Blog | Using Implied Volatility to Determine the Expected Range of a Stock
Using Implied Volatility to Determine the Expected Range of a Stock
When there is an event that is likely to impact the price of an equity (e. g. earnings; FDA ruling; new product release; etc.,) you will see an increase in the option pricing. This rise pricing is attributed to an increase in the option’s implied volatility. When the implied volatility is high, that means that the market anticipates a greater movement in the stock price.
I am often asked “How can I determine how far a stock is likely to move?” I will give you two methods to estimate the expected movement: one that requires a calculator and a quick and dirty way that you can do in your head.
First a little theory:
When we talk about historical volatility, we are measuring the zigzaggedness of a stock. How much did it zig and how much did it zag? If it zigged and zagged a lot, the historic volatility is high. In mathematical terms we determine the zigzaggedness by measuring the “standard deviation.”
Remember the bell curve or the normal distribution?
If a distribution is considered “normal,” 68% of the time, you will be within 1 standard deviation of the peak of that curve. For the most part, stock exhibit a normal distribution. (Actually, it’s a log normal distribution, but let’s keep this simple.)
The historic volatility is the movement that did occur. The implied volatility is the movement that is expected to occur in the future. When we are estimating future prices, we use the implied volatility.
Using the calculator:
The following calculation can be done to estimate a stocks potential movement:
(Stock price) x (Annualized Implied Volatility) x (Square Root of [days to expiration / 365]) = 1 standard deviation.
Take for example AAPL that is trading at $323.62 this morning. It has earnings next month. The current Implied Volatility is 31.6%. JAN options expire in 22 days, that would indicate that standard deviation is:
$323.62 x 31.6% x SQRT (22/365) = $25.11
That means that there is a 68% chance that AAPL will be between $298.51 and $348.73 in January expiration.
Quick and Dirty:
Now, for those of you that have not touched the square-root key since you took algebra back in high school, you can just use an options chain to get an estimate of where the stock could move. Find the price of the at-the-money straddle and the out-of-the-money strangle and add them together and divide by two. That is about equal to the 50% probability movement.
For AAPL this is the 320 straddle (320 call and put) and the 310/330 strangle (330 call and 310 put.) Add those together and you will get $30.58. If you divide that by two ($30.58 / 2 = $15.29) and add and subtract that from the current stock price, you get very close to 50% probability range. That means that AAPL has about a 50% chance of being at $308.33 and $338.91 by January expiration.
This method is a enough to make a mathematician cringe. However, it works out to be a pretty good “SWAG,” as they say.
Thoughts on what these numbers mean:
I don’t use these calculations as a price target to build a strategy. I use them more as a warning sign. Considering the potential movement, you may want to reconsider your delta neutral strategy (e. g. call calendar, put calendar, iron condor, etc.) which profits from a stagnant trend. While the premiums look good now, you might want to wait until after the event for that stagnant strategy. Or, you may be looking at a straddle or strangle knowing that there is going to be a big movement. If you look at those estimated moves, you’ll see that a straddle or strangle is probably not that attractive – especially when you consider the volatility crush that is likely to occur after the event.
For example the current JAN 320 straddle on AAPL. It’s pricing at $19.40. That means that you need the stock to move to at least $304.22 or $343.02 just to break even by expiration. Now, consider the estimates made above. Does this look like a trade that you’d want to be in? Unless you know something that the market doesn’t know – like the iPhone cures the common cold – you probably are best advised to avoid this sort of trade. (Another factor that must be considered is the dramatic changes in implied volatility. But, I will leave that for another post.)
Unless you have a strong sentiment, I believe the most prudent approach is to consider a protective strategy. For stock ownership, I believe that a collar trade is typically best strategy around earnings. It protects you from downward movement in the stock. Also, because there is a long and a short option, the impacts of volatility crush is mitigated. The art is picking the right options and that is what we teach you at OptionsAnimal.
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Abstracto
We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.
Article information
Dates First available in Project Euclid: 7 January 2009
Permanent link to this document http://projecteuclid. org/euclid. jap/1231340234
Digital Object Identifier doi:10.1239/jap/1231340234
Citación
Rogers, L. C. G.; Veraart, L. A. M. A stochastic volatility alternative to SABR. J. Appl. Probab. 45 (2008), no. 4, 1071--1085. doi:10.1239/jap/1231340234. http://projecteuclid. org/euclid. jap/1231340234.
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FX Derivatives Trader School
About this Book
An essential guide to real-world derivatives trading
FX Derivatives Trader School is the definitive guide to the technical and practical knowledge required for successful foreign exchange derivatives trading. Accessible in style and comprehensive in coverage, the book guides the reader through both basic and advanced derivative pricing and risk management topics.
The basics of financial markets and trading are covered, plus practical derivatives mathematics is introduced with reference to real-world trading and risk management. Derivative contracts are covered in detail from a trader's perspective using risk profiles and pricing under different derivative models. Analysis is approached generically to enable new products to be understood by breaking the risk into fundamental building blocks. To assist with learning, the book also contains Excel practicals which will deepen understanding and help build useful skills.
The book covers of a wide variety of topics, including:
Derivative exposures within risk management
Volatility surface construction
Implied volatility and correlation risk
Practical tips for students on trading internships and junior traders
Market analysis techniques
FX derivatives trading requires mathematical aptitude, risk management skill, and the ability to work quickly and accurately under pressure. There is a tremendous gap between option pricing formulas and the knowledge required to be a successful derivatives trader. FX Derivatives Trader School is unique in bridging that gap.
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Understanding the Implied Volatility Surface for Options on a Diversified Index
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Implied Volatility
Learn about implied volatility, how it effects trading strategies and download a spreadsheet.
Implied volatility is the volatility that matches the current price of an option, and represents current and future perceptions of market risk. This is in contrast to the normal definition of volatility, which is backwards-facing and is calculated from historical data (i. e. standard deviation of historical returns).
If traders expect the price of a stock to vary a lot, then its implied volatility (and Call and Put options) will trend upwards. Implied volatilities often exceed their historic counterparts prior to a major announcement (such as an earnings announcement or a merger), and tend to the mean afterwards. For example, if the market is enthusiastic about a specific stock (perhaps due to a great earnings report), then a Call option will be expensive. Accordingly, a covered Call is a good strategy.
Vega is rate of change in the value of an option given a 1% change in volatility. Hence knowing Vega prior to major announcements is essential in correctly pricing an option. Unless the price of a stock changes to reflect lower implied volatility, then puts/calls are expected to decline after a major announcement.
Some financial analysts consider implied volatility to be a price or value (rather than a statistical measure), given that it is directly derived from the transaction between a buyer-seller pair.
Calculate Implied Volatility with Excel
Excel’s Goal Seek can be used to backsolve for the volatility of a European Option (priced using Black-Scholes ) given the spot price, strike price, risk-free rate and time to expiration. An example is given in the spreadsheet below (scroll to the bottom for the download link), but let’s go through a worked example first.
Calculate the implied volatility of a European option with a
Spot Price of 490,
Strike Price of 470,
Risk-Free Rate of 0.033,
Expiry time of 0.08,
Call price of 30.
Step 1 . In the spreadsheet, enter the Spot price, Strike price, risk free rate and Expiry time. Also, enter an initial guess value for the volatility (this will give you an initial Call price that is refined in the next step)
Step 2 . Go to Data>What If Analysis>Goal Seek. Set the Call value to 30 (cell E5 in the spreadsheet) by changing the volatility (cell B8 in the spreadsheet)
You should find that volatility has been updated to 0.32 to reflect the desired Call price of 30.
Like the Free Spreadsheets?
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Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility
Received: 12 June 1997
In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define the arbitrage-free evolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair.
We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index.
You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.
Min Dai. Ling Tang. Xingye Yue. (2017) Calibration of stochastic volatility models: A Tikhonov regularization approach. Journal of Economic Dynamics and Control 64 . 66-81. Online publication date: 1-Mar-2017. [CrossRef]
Mathias Barkhagen. Jörgen Blomvall. Eckhard Platen. (2017) Recovering the real-world density and liquidity premia from option data. Quantitative Finance . 1-18. Online publication date: 26-Feb-2017. [CrossRef]
2017. Bibliography. Manufacturing and Managing Customer-Driven Derivatives, 521-530. [CrossRef]
Mathias Barkhagen. Jörgen Blomvall. (2017) Modeling and evaluation of the option book hedging problem using stochastic programming. Quantitative Finance 16 . 259-273. Online publication date: 1-Feb-2017. [CrossRef]
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2017. Bibliography. Equity Hybrid Derivatives, 313-321. [CrossRef]
Thomas Mazzoni. (2017) A GARCH Parameterization of the Volatility Surface. The Journal of Derivatives . 150819072208003. Online publication date: 19-Aug-2017. [CrossRef]
Thomas Mazzoni. (2017) A GARCH Parameterization of the Volatility Surface. The Journal of Derivatives 23 . 9-24. Online publication date: 1-Aug-2017. [CrossRef]
2017. Bibliography. Inside Volatility Filtering, 263-278. [CrossRef]
HIDEHARU FUNAHASHI. (2017) AN ANALYTICAL APPROXIMATION FOR EUROPEAN OPTION PRICES UNDER STOCHASTIC INTEREST RATES. International Journal of Theoretical and Applied Finance 18 :04. Online publication date: 1-Jun-2017. [Abstract | PDF (628 KB) | PDF Plus (640 KB) ]
STEVE ROSS. (2017) The Recovery Theorem. The Journal of Finance 70 :10.1111/jofi.2017.70.issue-2, 615-648. Online publication date: 1-Apr-2017. [CrossRef]
Youngrok Lee. Jaesung Lee. (2017) LOCAL VOLATILITY FOR QUANTO OPTION PRICES WITH STOCHASTIC INTEREST RATES. Korean Journal of Mathematics 23 . 81-91. Online publication date: 30-Mar-2017. [CrossRef]
Masaaki Kijima. Keiichi Tanaka. 2017. Alternatives to Black-Scholes Formulation in Finance Processes. Wiley StatsRef: Statistics Reference Online. [CrossRef]
Antonie Kotzé. Coenraad C. A. Labuschagne. Merell L. Nair. Nadine Padayachi. (2017) Arbitrage-free implied volatility surfaces for options on single stock futures. The North American Journal of Economics and Finance 26 . 380-399. Online publication date: 1-Dec-2017. [CrossRef]
Griselda Deelstra. Grégory Rayée. (2017) Local Volatility Pricing Models for Long-Dated FX Derivatives. Applied Mathematical Finance 20 . 380-402. Online publication date: 1-Sep-2017. [CrossRef]
2017. Bibliography. Volatility and Correlation, 805-812. [CrossRef]
Silvia Muzzioli. (2017) The Information Content of Option-Based Forecasts of Volatility: Evidence from the Italian Stock Market. Quarterly Journal of Finance 03 :01. Online publication date: 1-Mar-2017. [Abstract | PDF (1166 KB) | PDF Plus (817 KB) ]
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Lan Zhang. (2012) Implied and realized volatility: empirical model selection. Annals of Finance 8 . 259-275. Online publication date: 1-May-2012. [CrossRef]
V L Raju Chinthalapati. (2012) Volatility forecast in FX markets using evolutionary computing and heuristic techniques. 2012 IEEE Conference on Computational Intelligence for Financial Engineering & Economics (CIFEr) . 1-8. [CrossRef]
René Carmona. Sergey Nadtochiy. (2012) Tangent Lévy market models. Finance and Stochastics 16 . 63-104. Online publication date: 1-Jan-2012. [CrossRef]
Peter S. Karlsson. (2011) The Incompleteness Problem of the APT Model. Computational Economics 38 . 129-151. Online publication date: 1-Aug-2011. [CrossRef]
RENÉ CARMONA. SERGEY NADTOCHIY. (2011) TANGENT MODELS AS A MATHEMATICAL FRAMEWORK FOR DYNAMIC CALIBRATION. International Journal of Theoretical and Applied Finance 14 :01, 107-135. Online publication date: 1-Feb-2011. [Abstract | PDF (570 KB) | PDF Plus (459 KB) ]
Espen Gaarder Haug. Nassim Nicholas Taleb. (2011) Option traders use (very) sophisticated heuristics, never the Black–Scholes–Merton formula. Journal of Economic Behavior & Organization 77 . 97-106. Online publication date: 1-Feb-2011. [CrossRef]
Lech A. Grzelak. Cornelis W. Oosterlee. (2011) On the Heston Model with Stochastic Interest Rates. SIAM Journal on Financial Mathematics 2 . 255-286. Online publication date: 1-Jan-2011. [CrossRef]
Masaaki Kijima. Keiichi Tanaka. 2010. Alternatives to Black - Scholes Formulation in Finance. Encyclopedia of Statistical Sciences. [CrossRef]
Espen Gaarder Haug. Stein Frydenberg. Sjur Westgaard. (2010) Distribution and Statistical Behavior of Implied Volatilities. Business Valuation Review 29 . 186-199. Online publication date: 1-Nov-2010. [CrossRef]
Georgios Chalamandaris. Andrianos E. Tsekrekos. (2010) Predictable dynamics in implied volatility surfaces from OTC currency options. Journal of Banking & Finance 34 . 1175-1188. Online publication date: 1-Jun-2010. [CrossRef]
Valdo Durrleman. 2010. Implied Volatility: Market Models. Encyclopedia of Quantitative Finance. [CrossRef]
Yung-Ming Shiu. Ging-Ginq Pan. Shu-Hui Lin. Tu-Cheng Wu. (2010) Impact of Net Buying Pressure on Changes in Implied Volatility: Before and After the Onset of the Subprime Crisis . The Journal of Derivatives 17 . 54-66. Online publication date: 1-May-2010. [CrossRef]
H. Yin. Y. Wang. L. Qi. (2009) Shape-Preserving Interpolation and Smoothing for Options Market Implied Volatility. Journal of Optimization Theory and Applications 142 . 243-266. Online publication date: 1-Jul-2009. [CrossRef]
Yueh-Neng Lin. Chien-Hung Chang. (2009) VIX option pricing. Journal of Futures Markets 29 :10.1002/fut. v29:6, 523-543. Online publication date: 1-Jun-2009. [CrossRef]
Akihiko Takahashi. Akira Yamazaki. (2009) A new scheme for static hedging of European derivatives under stochastic volatility models. Journal of Futures Markets 29 :10.1002/fut. v29:5, 397-413. Online publication date: 1-May-2009. [CrossRef]
V. Moriggia. S. Muzzioli. C. Torricelli. (2009) On the no-arbitrage condition in option implied trees. European Journal of Operational Research 193 . 212-221. Online publication date: 1-Feb-2009. [CrossRef]
René Carmona. Sergey Nadtochiy. (2009) Local volatility dynamic models. Finance and Stochastics 13 . 1-48. Online publication date: 1-Jan-2009. [CrossRef]
Martin Schweizer. Johannes Wissel. (2008) Arbitrage-free market models for option prices: the multi-strike case. Finance and Stochastics 12 . 469-505. Online publication date: 1-Oct-2008. [CrossRef]
S. Muzzioli. H. Reynaerts. (2008) American option pricing with imprecise risk-neutral probabilities. International Journal of Approximate Reasoning 49 . 140-147. Online publication date: 1-Sep-2008. [CrossRef]
Carol Alexander. Leonardo M. Nogueira§. (2007) Model-free price hedge ratios for homogeneous claims on tradable assets. Quantitative Finance 7 . 473-479. Online publication date: 1-Oct-2007. [CrossRef]
Carol Alexander. Leonardo M. Nogueira. (2007) Model-free hedge ratios and scale-invariant models. Journal of Banking & Finance 31 . 1839-1861. Online publication date: 1-Jun-2007. [CrossRef]
Wael Bahsoun. Paweł Góra. Silvia Mayoral. Manuel Morales. (2007) Random dynamics and finance: constructing implied binomial trees from a predetermined stationary density. Applied Stochastic Models in Business and Industry 23 :10.1002/asmb. v23:3, 181-212. Online publication date: 1-May-2007. [CrossRef]
Mark H. A. Davis. David G. Hobson. (2007) THE RANGE OF TRADED OPTION PRICES. Mathematical Finance 17 :10.1111/mafi.2007.17.issue-1, 1-14. Online publication date: 1-Jan-2007. [CrossRef]
Theo Darsinos. Stephen Satchell. 2007. Bayesian forecasting of options prices. Forecasting Expected Returns in the Financial Markets, 151-175. [CrossRef]
Rolf Poulsen. (2006) Barrier options and their static hedges: simple derivations and extensions. Quantitative Finance 6 . 327-335. Online publication date: 1-Aug-2006. [CrossRef]
Marco Fabio Delzio. (2006) Pricing credit risk through equity options calibration. The Journal of Risk Finance 7 . 372-385. Online publication date: 1-Aug-2006. [CrossRef]
Morten Nalholm. Rolf Poulsen. (2006) Static hedging and model risk for barrier options. Journal of Futures Markets 26 :10.1002/fut. v26:5, 449-463. Online publication date: 1-May-2006. [CrossRef]
CHARILAOS E. LINARAS. GEORGE SKIADOPOULOS. (2005) IMPLIED VOLATILITY TREES AND PRICING PERFORMANCE: EVIDENCE FROM THE S&P 100 OPTIONS. International Journal of Theoretical and Applied Finance 08 :08, 1085-1106. Online publication date: 1-Dec-2005. [Abstract | PDF (604 KB) | PDF Plus (544 KB) ]
Graeme West. (2005) Calibration of the SABR Model in Illiquid Markets. Applied Mathematical Finance 12 . 371-385. Online publication date: 1-Dec-2005. [CrossRef]
Truc Le. (2005) Stochastic market volatility models. Applied Financial Economics Letters 1 . 177-188. Online publication date: 1-May-2005. [CrossRef]
George J. Jiang. Yisong S. Tian. (2005) The Model-Free Implied Volatility and Its Information Content. Review of Financial Studies 18 . 1305-1342. Online publication date: 1-Jan-2005. [CrossRef]
Henri Berestycki. J r me Busca. Igor Florent. (2004) Computing the implied volatility in stochastic volatility models. Communications on Pure and Applied Mathematics 57 :10.1002/cpa. v57:10, 1352-1373. Online publication date: 1-Oct-2004. [CrossRef]
Nikolaos Panigirtzoglou. George Skiadopoulos. (2004) A new approach to modeling the dynamics of implied distributions: Theory and evidence from the S&P 500 options. Journal of Banking & Finance 28 . 1499-1520. Online publication date: 1-Jul-2004. [CrossRef]
Kirill Ilinski. Oleg Soloviev. (2004) Stochastic volatility membrane. Wilmott 2004 :10.1002/wilm. v2004:3, 74-81. Online publication date: 1-May-2004. [CrossRef]
Damiano Brigo. Fabio Mercurio. Giulio Sartorelli. (2003) Alternative asset-price dynamics and volatility smile. Quantitative Finance 3 . 173-183. Online publication date: 1-Jun-2003. [CrossRef]
Damiano Brigo. Fabio Mercurio. (2003) Analytical pricing of the smile in a forward LIBOR market model. Quantitative Finance 3 . 15-27. Online publication date: 1-Feb-2003. [CrossRef]
S Cr pey. (2003) Calibration of the local volatility in a trinomial tree using Tikhonov regularization. Inverse Problems 19 . 91-127. Online publication date: 1-Feb-2003. [CrossRef]
DAMIANO BRIGO. FABIO MERCURIO. (2002) LOGNORMAL-MIXTURE DYNAMICS AND CALIBRATION TO MARKET VOLATILITY SMILES. International Journal of Theoretical and Applied Finance 05 :04, 427-446. Online publication date: 1-Jun-2002. [Abstract | PDF (306 KB) | PDF Plus (370 KB) ]
P Balland. (2002) Deterministic implied volatility models. Quantitative Finance 2 . 31-44. Online publication date: 1-Feb-2002. [CrossRef]
Rama Cont. José da Fonseca. (2002) Dynamics of implied volatility surfaces. Quantitative Finance 2 . 45-60. Online publication date: 1-Feb-2002. [CrossRef]
Claude Bardos. Raphaël Douady. Andrei Fursikov. (2002) Static Hedging of Barrier Options with a Smile: An Inverse Problem. ESAIM: Control, Optimisation and Calculus of Variations 8 . 127-142. Online publication date: 1-Jan-2002. [CrossRef]
Thierry Ané. Chiraz Labidi. (2001) Implied volatility surfaces and market activity over time. Journal of Economics and Finance 25 . 259-275. Online publication date: 1-Sep-2001. [CrossRef]
KARL STROBL. (2001) ON THE CONSISTENCY OF THE DETERMINISTIC LOCAL VOLATILITY FUNCTION MODEL ('IMPLIED TREE'). International Journal of Theoretical and Applied Finance 04 :03, 545-565. Online publication date: 1-Jun-2001. [Abstract | PDF (349 KB) | PDF Plus (300 KB) ]
ROGER W. LEE. (2001) IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY. International Journal of Theoretical and Applied Finance 04 :01, 45-89. Online publication date: 1-Feb-2001. [Abstract | PDF (521 KB) | PDF Plus (590 KB) ]
Mark Britten-Jones. Anthony Neuberger. (2000) Option Prices, Implied Price Processes, and Stochastic Volatility. The Journal of Finance 55 . 839-866. Online publication date: 1-Apr-2000. [CrossRef]
Jens Carsten Jackwerth. (1999) Option-Implied Risk-Neutral Distributions and Implied Binomial Trees. The Journal of Derivatives 7 . 66-82. Online publication date: 1-Jan-1999. [CrossRef]
Forex Pdf Article:
Is There A Best Time To Trade Forex Market? By Caterina Christakos The answer to the question," Is there a best time to trade the forex?" depends on your objectives. If you are trading the based on earning a higher rate of interest then Wednesdays are the best day to trade the forex. You earn triple the interest on your currency trades.
The market, as you know, is a 24 hour market.
Forex trading hours, trading time:
New York opens 8:00 am to 5:00 pm EST Tokyo opens 7:00 pm to 4:00 am EST Sydney opens 5:00 pm to 2:00 am EST London opens 3:00 am to 12:00 noon EST
As you can see there is an overlap in trading times.
New York and London - 8:00 am - 12:00 noon EST Sydney / Tokyo - 7:00 pm - 2:00 am EST London /Tokyo - 3:00 am - 4:00am EST
As trading pairs overlap, they become more active. If you are day trading the these would be the times of greatest volatility.
that the market is unpredictable. As we have just seen the stock market ( February - March 2007) can affect the volatility of the market.
Have your money and trading rules in mind before entering into any trade. Trade a percentage of your account that you are comfortable with and that will not leave you with a margin call if the market does take a major dip.
Have your exit strategies in mind before entering your trades as well. Don't be greedy. There are fortunes to be made in the market but fortunes have also been lost here.
The key to success in this or any trading market is to know what level of risk that you are comfortable with and to trade with your money rules in mind and not your ego.
Trade to make a profit not to get high or pat yourself on the back for your own brilliance at the occasional slam dunk.
Search for more information about Forex Pdf
Calculating Implied Volatility in Excel
The Black-Scholes option pricing formula can’t be deconstructed to determine a direct formula for implied volatility. However, if you know the option’s price and all the remaining parameters (underlying price, strike price, interest rate, dividend yield, and time to expiration), you can use the Goal Seek feature in Excel to find it. This page explains how to do it in the Black-Scholes Calculator (but the logic is the same if you do it on your own and prepare all the Black-Scholes model formulas yourself).
I will illustrate the Excel calculation of implied volatility step-by-step on the example below.
Ejemplo
You want to find implied volatility of a call option with strike price of 55 and 18 calendar days to expiration. The risk free interest rate is 1%; the underlying stock’s continuously compounded dividend yield is 2%. The underlying stock is currently trading at 53.20 and the option is trading at 1.40.
Setting the Input Parameters
First, you must set all the parameters that enter option price calculation:
Enter 53.20 in cell C4 (Underlying Price)
Enter 55 in cell C6 (Strike Price)
Cell C8 contains volatility, which you don’t know. Just enter something (for example 50%).
Enter 1% in cell C10 (Interest Rate)
Enter 2% in cell C12 (Dividend Yield) – if the underlying pays no dividend, enter zero or leave this cell blank.
Enter 18 in cell C19 (Time to Expiration) and select “Calendar days” in the combo in cell C20. If you don’t know the number of days, but you know the expiration date of the option, enter the dates in cells C15 and C17 (if both these cells contain numbers, they will override the input in cell C19). You may also enter exact times in cells C16 and C18 if you want to be very precise.
Now you have entered all the parameters and the resulting option price appears in cell H4 (or H6 if you are working with a put option). Using the values in our example, it is 1.59 dollars per share.
Unless you were very lucky, it is not equal to the actual price at which the option is trading at the moment (in our example 1.40). The reason is the volatility parameter, where you have entered a number you just guessed.
Note: Do not type any numbers in cells H4 and H6 (the resulting option prices). These cells contain formulas and if you overwrite them, the spreadsheet will not work correctly.
Trial and Error Approach
Now you can try to find the implied volatility by trial and error by entering different values in cell C8 . If the real option price is lower than your result (as in our example), try lower volatility, and vice versa. For example, you can try to enter 45% into cell C8 and get option price of 1.36. So you try 47% and get 1.45… y así. As soon you get close enough to the real option price (depending on your desired level of accuracy), you are done. The implied volatility is in cell C8.
Using Excel Goal Seek
The Goal Seek feature in Excel does exactly the same thing, only the computer is able to perform this trial and error exercise in split of a second and get a very accurate result immediately.
Once you have the input parameters set, go to Excel main menu and select Data . Data Tools . What-If Analysis . Goal Seek (in Excel 2010 – the path may be slightly different in other versions).
The Goal Seek window pops up and asks you to enter three inputs:
“Set cell:” – the cell where the resulting option price is calculated – enter H4 if you are trying to find implied volatility of a call (our example), or H6 for a put.
“To value:” – the option’s price. In our example enter 1.40.
“By changing cell:” – the cell which contains the input that you want to find – in our example the cell with volatility input, C8.
Now press OK and the desired implied volatility appears in cell C8 (45.83% in our example). At the same time, the option‘s price (1.40 in our example) should appear in cell H4 (or H6 if it was a put). You can also see the option’s delta, gamma, theta, vega, and rho to the right of option’s price in cells J4 to N4 (or J6 to N6).
Black-Scholes Calculator + PDF Guide
This tutorial is part of the PDF Guide that comes with the Black-Scholes Calculator. You can see more information about all features, calculations, and guide contents here.
Frustrated with all those useless trend indicators that just don't work?
This Volatility Hypertrend Indicator Could Be The Most Accurate Trend Indicator The Industry Has Ever Seen.
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According to our statistical tests, this is the most accurate trend indicator. And it achieves that by incorporating market volatility in its algorithm to increase accuracy, and eliminate false signals.
Does this sound familiar?
Your traditional trend indicator (you know, like MACD or ADX) tells you a new trend has established.
So you enter a trade. But the price is not really going anywhere. It's just moving sideways, back and forth. toying with you.
Or even worse, many times the price quickly reverses against my position as soon as I place a trade. And I take an embarrassing loss from this stupid false signal.
You see, here's the thing:
False signals are INEVITABLE with traditional trend indicators.
They apply the same calculation no matter which currency pair you trade, and no matter which time frame you use.
And they don't take into account the volatility of a particular currency pair, at a particular time.
I think it's a fatal mistake.
Can you imagine using the same method when you trade a volatile pair (like GBP/USD which has a large daily range), as when you trade a pair like EUR/GBP which has a much smaller range?
That's why you need a new kind of trend indicator that incorporates market volatility in its algorithm.
Volatility Hypertrend Indicator
This "Volatility Hypertrend" indicator is the first indicator that uses both a Volatility algorithm and a Trend algorithm to identify a trend.
That's the secret to its accuracy.
The indicator clearly displays market trends on your charts, shows you exactly where possible reversals occurs.
And gives you audio & visual alerts when the trend changes direction.
Let's take a look at how the indicator works:
See how well the Volatility Hypertrend indicator determines the market trend?
In this example, even if you used a super simple strategy like buying when the indicator signals an uptrend. and selling when the indicator signals a downtrend. you would have already generated 1,108 pips in profit.
Not too shabby, right?
Let's take a look at another case study. this time on USD/CAD daily time frame:
As you can see, Volatility Hypertrend indicator works exceedingly well at signaling when the market changes direction.
Más. because it takes market volatility into account.
It helps you avoid false signals when sideways movement or market correction occurs.
Let's walk through another case study:
In this case study, because false signals are eliminated, the indicator signals 4 possible market reversals. And if you just followed these signals, nothing more nothing less, you would have captured most of this huge downtrend.
On top of that, you would have had a few small winners from these market corrections.
Gotta love these simple wins!
One more example: (USD/CHF daily chart)
If Volatility Hypertrend indicator gave you these signals, would you be able to profit from these clear trends?
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Here's ANOTHER great way to use Volatility Hypertrend indicator.
Did you know that the Volatility Hypertrend indicator also gives you the exact price where you should set your stop loss?
Just hover your mouse on the Volatility Hypertrend indicator line, and your suggested stop loss will show up. Me gusta esto:
By the way, the Volatility Hypertrend indicator will generate an ALERT for you as soon as a possible trend reversal happens.
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Last time I did something like this, 2,000 copies were grabbed in a few days. And that just overwhelmed me.
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Gobierno de los Estados Unidos Exención de Responsabilidad - Commodity Futures Trading Commission El comercio de futuros y opciones tiene grandes recompensas potenciales, pero también un gran riesgo potencial. Debe ser consciente de los riesgos y estar dispuesto a aceptarlos para invertir en los mercados de futuros y opciones. No negocie con dinero que no puede permitirse perder. Esto no es ni una solicitud ni una oferta de compra / venta de futuros u opciones. No se está haciendo ninguna representación de que cualquier cuenta tenga o sea probable obtener ganancias o pérdidas similares a las discutidas en este sitio web. El desempeño pasado de cualquier sistema o metodología comercial no es necesariamente indicativo de resultados futuros.
CFTC REGLA 4.41 - LOS RESULTADOS DE RENDIMIENTO HIPOTÉTICOS O SIMULADOS TIENEN CIERTAS LIMITACIONES. DESCONOCIDO UN REGISTRO DE RENDIMIENTO REAL, LOS RESULTADOS SIMULADOS NO REPRESENTAN COMERCIO REAL. TAMBIÉN, DADO QUE LOS COMERCIOS NO HAN SIDO EJECUTADOS, LOS RESULTADOS PUEDEN TENERSE COMPARTIDOS POR EL IMPACTO, SI CUALQUIERA, DE CIERTOS FACTORES DE MERCADO, COMO LA FALTA DE LIQUIDEZ. LOS PROGRAMAS DE COMERCIO SIMULADOS EN GENERAL ESTÁN SUJETOS AL FACTOR DE QUE SEAN DISEÑADOS CON EL BENEFICIO DE HINDSIGHT. NO SE HACE NINGUNA REPRESENTACIÓN QUE CUALQUIER CUENTA TENDRÁ O ES POSIBLE PARA LOGRAR GANANCIAS O PÉRDIDAS SIMILARES A LOS MOSTRADOS.
Use of any of this information is entirely at your own risk, for which we will not be liable. Neither we nor any third parties provide any warranty or guarantee as to the accuracy, timeliness, performance, completeness or suitability of the information and content found or offered in the material for any particular purpose. You acknowledge that such information and materials may contain inaccuracies or errors and we expressly exclude liability for any such inaccuracies or errors to the fullest extent permitted by law. All information exists for nothing other than entertainment and general educational purposes. We are not registered trading advisors.
3 Ways Commodity Options Trading Differs from Stock Options
I'm a huge fan of trading commodity options.
And you should be, too.
But there are some nuances when trading commodity options that you have to consider if you're going to move some capital away from equities.
These 3 differences are the most important concepts to understand, as they can potentially change the way you trade these instruments.
Fear is In Both Directions
If there is one thing to learn about options is that each contract will have a different implied volatility. You can visualize implied volatility over various strikes by looking at the volatility skew.
Below is a picture from LiveVol showing the volatility skew for SPY June Options:
Notice that as we go lower in strike, the implied volatility on each contract rises. This is because option traders are willing to pay up for "tail-risk" protection, and most hedgers in equities are fearful of downside.
Compare that with the volatility skew for GLD June Options:
Instead of a "skew" we now have a "smile." What is going on here?
It comes down to the perception of risk . Equity investors are fearful of downside in equities. But in commodities like gold, oil, soybeans, and currencies the perception of risk is bi-directional.
That means the tail risk can be on either side.
Think about oil-- if we saw a $20 move in oil to the upside in a very short amount of time, that would have significant consequences across various assets.
So when hedgers and speculators come out to commodity options, they fear strong moves in either direction.
This changes the strategy set used in commodity options trading-- iron condors become more attractive, as do ratio sales after extreme moves.
Commodities Have Different Event Based Risk
Single stock equities can be driven by upgrades, downgrades, earnings, FDA events, insider selling, holding updates, institutional rebalancing, intermarket correlation, same store sales. the list can keep going.
This heightened risk produces higher potential reward-- and for those that want to get more conservative, trading indexes or index futures can mitigate that risk.
With commodity options, the risks that drive movement are quite different than what drives equities. It could be based off supply reports or interest rate changes by central banks.
Because the risks are different, it can give you a way to diversify your trades against different risks. This can be crucial when finding the best trades.
Commodity Option Traders Are a Different Breed
Remember, it comes down to the perception of risk.
Why is risk bi-directional?
Because the motivations in the commodities market are completely different than stocks.
Joe farmer needs to sell his soybeans. Spacely Sprocket company needs to hedge their Euro risk. ZeroHedge has to buy more silver to combat the manipulators.
Contrast that with stocks-- for 95% of the population, investing is the key term. They look to buy stock in companies.
Contrast that to gold and oil: there's no cash flow from these. From a structural standpoint, they aren't "investments."
Because the needs of these markets are different, the demands of these markets are different.
What's going on right now.
I see two possibilities heading into the summer months . Either event-based risk (Eurozone/China) makes a comeback or it doesn't. If we get the first scenario, then correlations will ratchet up among stocks and it will be a macro game again. If the second comes along, then summer volatility and liquidity in equities will dwindle.
Either way, commodity options trading is definitly coming back into my trading arsenal for the next few months.
Steven Place is the founder and head trader at investingwithoptions. com/
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Asset Allocation Implications of the Global Volatility Premium
The authors examined the role of volatility premiums in institutional investment portfolios. They began by defining and calculating standardized returns to volatility exposure for a variety of global asset markets. They found that shorting volatility offers not only a very high and statistically significant Sharpe ratio of approximately 1.0 but also substantial tail risk. Although classic diversification benefits are limited, the authors show that modest allocations to short volatility exposure could have enhanced long-term returns, in one case increasing the portfolio’s combined Sharpe ratio by 12%.
Editor’s notes: The authors may have a commercial interest in the topics discussed in this article. This article was reviewed and accepted by Executive Editor Robert Litterman.
Authors’ note: The views and opinions expressed herein are those of the authors and do not reflect the views of Goldman Sachs. The backtests and analysis described herein are provided for educational purposes in reliance on past market data with the benefit of hindsight and do not reflect actual results. If any assumptions used do not prove to be true, results may vary substantially. Our research does not take into account specific investment objectives, investor guidelines, or restrictions. Investors must also consider suitability, liquidity needs, and investment objectives when determining appropriate asset allocation. All swap and swaption data are courtesy of J. P. Morgan Research, copyright 2017.
William Fallon is chief investment officer at Goldman Sachs Asset Management in New York City.
James Park is a portfolio manager in the Macro Quantitative Investment Strategies group at Goldman Sachs Asset Management in New York City.
Danny Yu is a portfolio manager in the Macro Quantitative Investment Strategies group at Goldman Sachs Asset Management in New York City.
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The Mathematics of Finance: Modeling and Hedging
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Victor Goodman; Joseph Stampfli
This book is ideally suited for an introductory undergraduate course on financial engineering. It explains the basic concepts of financial derivatives, including put and call options, as well as more complex derivatives such as barrier options and options on futures contracts. Both discrete and continuous models of market behavior are developed in this book. In particular, the analysis of option prices developed by Black and Scholes is explained in a self-contained way, using both the probabilistic Brownian Motion method and the analytical differential equations method.
The book begins with binomial stock price models, moves on to multistage models, then to the Cox–Ross–Rubinstein option pricing process, and then to the Black–Scholes formula. Other topics presented include Zero Coupon Bonds, forward rates, the yield curve, and several bond price models. The book continues with foreign exchange models and the Keynes Interest Rate Parity Formula, and concludes with the study of country risk, a topic not inappropriate for the times.
In addition to theoretical results, numerical models are presented in much detail. Each of the eleven chapters includes a variety of exercises.
An instructor's manual for this title is available electronically. Please send email to textbooks@ams. org for more information.
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The Mathematics of Finance: Modeling and Hedging
Base Product Code Keyword List: amstext ; AMSTEXT ; amstext/7 ; AMSTEXT/7 ; amstext-7 ; AMSTEXT-7
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Title (HTML): The Mathematics of Finance: Modeling and Hedging
Author(s) (Product display): Victor Goodman ; Joseph Stampfli
Affiliation(s) (HTML): Indiana University, Bloomington, IN ; Indiana University, Bloomington, IN
This book is ideally suited for an introductory undergraduate course on financial engineering. It explains the basic concepts of financial derivatives, including put and call options, as well as more complex derivatives such as barrier options and options on futures contracts. Both discrete and continuous models of market behavior are developed in this book. In particular, the analysis of option prices developed by Black and Scholes is explained in a self-contained way, using both the probabilistic Brownian Motion method and the analytical differential equations method.
The book begins with binomial stock price models, moves on to multistage models, then to the Cox–Ross–Rubinstein option pricing process, and then to the Black–Scholes formula. Other topics presented include Zero Coupon Bonds, forward rates, the yield curve, and several bond price models. The book continues with foreign exchange models and the Keynes Interest Rate Parity Formula, and concludes with the study of country risk, a topic not inappropriate for the times.
In addition to theoretical results, numerical models are presented in much detail. Each of the eleven chapters includes a variety of exercises.
An instructor's manual for this title is available electronically. Please send email to textbooks@ams. org for more information.
Book Series Name: Pure and Applied Undergraduate Texts
Publication Month and Year: 2009-03-10
Financial Humor
Riesgo: DailyForex no se hace responsable de ninguna pérdida o daño resultante de la confianza en la información contenida en este sitio web, incluyendo noticias de mercado, análisis, señales comerciales y revisiones de corredores de Forex. Los datos contenidos en este sitio web no son necesariamente en tiempo real ni precisos, y los análisis son opiniones del autor y no representan las recomendaciones de DailyForex ni de sus empleados. El comercio de divisas en margen conlleva un alto riesgo y no es adecuado para todos los inversores. Como producto apalancado, las pérdidas pueden exceder los depósitos iniciales y el capital está en riesgo. Antes de decidir negociar Forex o cualquier otro instrumento financiero, debe considerar cuidadosamente sus objetivos de inversión, nivel de experiencia y apetito por el riesgo.
Riesgo: DailyForex no se hace responsable de ninguna pérdida o daño resultante de la confianza en la información contenida en este sitio web, incluyendo noticias de mercado, análisis, señales comerciales y revisiones de corredores de Forex. Los datos contenidos en este sitio web no son necesariamente en tiempo real ni precisos, y los análisis son opiniones del autor y no representan las recomendaciones de DailyForex ni de sus empleados. El comercio de divisas en margen conlleva un alto riesgo y no es adecuado para todos los inversores. Como producto apalancado, las pérdidas pueden exceder los depósitos iniciales y el capital está en riesgo. Antes de decidir negociar Forex o cualquier otro instrumento financiero, debe considerar cuidadosamente sus objetivos de inversión, nivel de experiencia y apetito por el riesgo.
Division of Mathematical Statistic, Centre for Mathematical Sciences, Lund University, P. O. Box 118, 22100 Lund, Sweden
Received 7 September 2009; Accepted 13 February 2010
Academic Editor: Henry Schellhorn
Copyright © 2010 Erik Lindström. This is an open access article distributed under the Creative Commons Attribution License. which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstracto
Financial markets are complex processes where investors interact to set prices. We present a framework for option valuation under imperfect information, taking risk neutral parameter uncertainty into account. The framework is a direct generalization of the existing valuation methodology. Many investors base their decisions on mathematical models that have been calibrated to market prices. We argue that the calibration process introduces a source of uncertainty that needs to be taken into account. The models and parameters used may differ to such extent that one investor may find an option underpriced; whereas another investor may find the very same option overpriced. This problem is not taken into account by any of the standard models. The paper is concluded by presenting simulations and an empirical study on FX options, where we demonstrate improved predictive performance (in sample and out of sample) using this framework.
1. Introducción
Mathematical models are used in the financial industry for prediction and risk management. The quality of the models is crucial—during summer 2007, media reported
We are seeing things that were 25-standard deviation events, several days in a row. David Viniar, Goldman Sachs CFO .
The Goldman Sachs GEOfund lost 30% of its value in a week, due to rare events. Assessing and controlling these risks is of vital interest to avoid unpleasant surprises. A 25-standard deviation event should almost never occur if the data generating process is Gaussian (the probability of an event of this size or larger is roughly
) but will occur from time to time if the model is heavy tailed.
Mathematical models are either parametric or nonparametric. Parametric models are the dominating approach, as these are easier to analyze and easier to fit to data. A limitation (and simultaneously the strength) of the parametric models is their limited flexibility, resulting in low variance and some bias; whereas nonparametric models are flexible and less biased but often poor (highly variable) predictors (cf. [1 ]).
Popular parametric option valuation models include the Black and Scholes model [2 ], the Merton jump diffusion [3 ], the Heston stochastic volatility [4 ], the Bates stochastic volatility jump diffusion [5 ], exponential Lévy processes such as Variance Gamma [6 ], Normal Inverse Gaussian [7 ], CGMY [8 ], and stochastic intensity models [9. 10 ].
However, it was claimed by [11 ] that even advanced models cannot explain all features in data (in the volatility surface), suggesting that additional or alternative modeling is needed. This view is shared by [12 ] where it is concluded that transaction costs are the key determinant of the curvature of the smile.
Parametric models are calibrated to data by estimating the parameters, often by minimizing some loss function, (cf. [13 ]) or through nonlinear Kalman filters; see [14 ]. It is easily argued that inaccurate calibration methods can cause problems. Inefficient estimators imply that another estimator would give better estimates. Oversimplified models will also cause problems, directly by providing biased forecasts and indirectly by having large parameter uncertainty due to large residual variance while over parameterized model will overfit data. Estimators such as Gaussian Maximum Likelihood estimators minimize the mean square prediction errors, trading bias for variance to achieve this.
Several calibration methods are primarily designed to estimate the volatility, although other parameters may also be of interest. Real-world calibration is complex as several subjective choices need to be made. Two things are needed, an estimator and a set of data . Starting with the estimators, we have to choose between the following. (i) Historical volatility, which is the standard (MLE) volatility estimator, scaled to account for the sampling frequency. Statistical theory (Cramer-Rao bounds, etc.) suggests that this should be the optimal estimator, given that the volatility is constant over time. More recent variation on this theme includes realized volatility and Bipower variation (cf. [15 ]). (ii) Time series models, such as ARCH/GARCH models [16 ], stochastic volatility models, or EWMA filters. These provide volatility forecasts that capture temporal variations in the volatility. (iii) Implied volatility is a common name for estimating the volatility from quoted options rather than from the underlying asset. The simplest implied volatility estimator is found by inverting the Black and Scholes formula, while other estimators such as VIX use a combination of prices having different moneyness and time to maturity to estimate the volatility.
Studies have shown (cf. [17 ]) that implied estimators often outperform all other estimators, even though recent realized volatility estimators are increasing efficient and may also provide good estimates. Reference [17 ] explained their findings by the fact that implied estimators look forward in time; whereas other estimators extrapolate from historical data. Another explanation is the higher quality of the data: a single option provides a reasonable estimate of the volatility while historical estimators require large data sets to provide good estimates.
The purpose of the estimators is also important as most estimators will only estimate either the objective measure
or the risk-neutral measure
and both are needed when hedging options in incomplete markets.
Another, but related problem, is selection of data. Several factors will influence the result. (i) The sampling frequency can be of paramount importance! Data sampled at higher frequencies should in theory give better estimates, but market microstructure (e. g. ask-bid spreads) tends to invalidate some of the gain. A related problem is that different time scales have different dependence structures. The correlation structure in high frequency data is sometimes claimed to be similar to long range dependence, while correlation structure in daily or weekly data is ordinary (e. g. exponentially decaying). (ii) The size of the estimation window can influence the results. Restricting the data set to recent data will lead to noisy estimates, while including too much historical data leads to bias and difficulties to track market variations.
It is highly unlikely, taking different estimators and data sets into consideration, that all investors are using identical estimates, thereby causing the “market parameters” to be unknown.
The purpose of this paper is to value options under parameter uncertainty. Reference [18 ] studies model uncertainty, which is related to parameter uncertainty. The primary purpose of their paper is not to price the model risk, but rather quantify the size of the risk. We believe that it is of importance to value the parameter uncertainty, for example, when computing hedges.
Valuation of options under parameter uncertainty was treated in [19. 20 ]. Both papers use a Bayesian framework to compute the posterior distribution of prices. However, their approach is purely statistical (the expectation is taken over the objective, distribution) and is not based on financial theory ( distribution). Averaging over the - distribution when the parameters should be used could easily result in biased. Still, their work is important as model averaging usually improves predictive performance (cf. [21 ]).
Reference [22 ] introduces stochastic parameters as a method of improving the fit of basic models. Their resulting valuation formulas are similar to what we derive for simple (Black and Scholes-like) models.
It is organized as follows. In Section 2. we review the basics of risk-neutral valuation framework. In Section 3. we proceed by suggesting a modification to the standard risk neutral valuation, and Section 4 presents some simulations in this framework. Section 5 provides an empirical study on FX options, and Section 6 concludes the paper.
2. Valuation of Options
The basis for valuation of contingent claims is the risk neutral valuation formula; see [23 ]. Dejar
An analytic multi-currency model with stochastic volatility and stochastic interest rates
We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed effciently through the FFT methodology thanks to the affinity of the model Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fit simultaneously FX implied volatilities while being coherent with interest rate products.
Si experimenta problemas al descargar un archivo, compruebe si tiene la aplicación adecuada para verla primero. En caso de problemas adicionales, lea la página de ayuda de IDEAS. Tenga en cuenta que estos archivos no están en el sitio IDEAS. Por favor sea paciente ya que los archivos pueden ser grandes.
References listed on IDEAS Please report citation or reference errors to. o. Si usted es el autor registrado del trabajo citado, inicie sesión en su perfil de servicio de RePEc Author. click on "citations" and make appropriate adjustments.
Garman, Mark B. & Kohlhagen, Steven W. 1983. " Foreign currency option values ," Journal of International Money and Finance. Elsevier, vol. 2(3), pages 231-237, December.
Griselda Deelstra & Grégory Rayée, 2017. " Local Volatility Pricing Models for Long-Dated FX Derivatives ," Applied Mathematical Finance. Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," HSC Research Reports HSC/10/02, Hugo Steinhaus Center, Wroclaw University of Technology.
Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," Papers 1010.1617, arXiv. org.
Agnieszka Janek & Tino Kluge & Rafał Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," SFB 649 Discussion Papers SFB649DP2010-047, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
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Publicaciones recientes de Christian Landmark: Christian Landmark
Face-Lift (Rhytidectomy)
Surgery Overview
A face-lift is the most extensive way to remove or reduce the appearance of wrinkles and sagging of the face caused by age. In a traditional face-lift, the skin is literally lifted off the face so that the skin and the tissues beneath can be tightened and the skin can be repositioned smoothly over the face.
For the procedure, you are either given general anesthesia or a sedative through an intravenous line and local anesthesia to numb your skin. Next, the surgeon makes an incision that starts in the temple area and circles around the front of the ear. The skin is raised, and the muscle and tissue underneath is tightened. The surgeon may remove some fat and skin. The skin is then redraped over the face and the incision is sutured. The incision usually falls along the hairline or in a place where the skin would naturally crease so that it does not show after the surgery.
Some people are able to have a limited-incision face-lift. This surgery uses shorter incisions at the temple and close to the ear. Sometimes an incision is made within the lower eyelid or under the upper lip.
A neck lift can tighten sagging jowls and loose skin under the chin. The incision starts in front of the ear lobe and goes behind the ear to the lower scalp.
The surgery usually takes several hours. You may be able to go home that day. But people sometimes spend one night in the hospital.
What To Expect After Surgery
Your face will be bandaged after the surgery. The dressings are usually removed 1 to 2 days later. If a drainage tube has been placed (usually behind the ear), it will also be removed 1 to 2 days after the surgery. Your doctor will remove your stitches in 5 to 10 days.
Your doctor may prescribe medicines to relieve pain after the surgery. Expect to have swelling and bruising of the face. Cold compresses can help relieve these side effects. Your doctor may instruct you to keep your head elevated and still as much as possible.
It is important to avoid smoking and even second-hand smoke for 2 to 4 weeks before and after surgery. Tobacco smoke increases the risk for skin and tissue death and will delay your face's healing process and make scarring worse.
Most people can return to their normal activities 2 to 3 weeks after a face-lift.
At first your face will feel stiff and will probably look and feel strange to you. This is normal, but it is important to be prepared for it.
Numbness of the skin may last for months after the surgery. Your skin may feel rough and dry for a few months. Men sometimes have to shave in new places because the skin has been rearranged, but laser hair removal or electrolysis can be used for beard hairs that have shifted to a new position.
Why It Is Done
Face-lifts are done to make an older face look younger by eliminating wrinkles. lifting sagging muscles, and tightening the skin.
How Well It Works
Having a face-lift can make your face appear younger and healthier. Your face will continue to age, but a face-lift does indeed "take years off" your face. For some people, this may increase self-confidence and reduce anxiety over growing older.
A face-lift can reduce signs of aging to a great extent. But it cannot reverse sun damage to the skin or remove all facial wrinkles around the eyes. below the nose, and around the lips. For best results, you may want to have a face-lift and then treat any skin damage.
The effectiveness and safety of your face-lift surgery depends heavily on the skill of your surgeon.
riesgos
Problems that may be caused by having a face-lift include:
Reactions to the anesthesia.
Bleeding under the skin.
Infection.
Damage to the nerves that supply the muscles of the face. This can cause paralysis or spasm in the face, but the effects are usually temporary.
Numbness (in areas of your face) that may last 6 months to a year.
Hair loss (alopecia ).
Tissue loss.
Scarring.
Blood clots in large veins traveling up to the heart and lungs (pulmonary embolism ). This is not common.
As with all cosmetic procedures, there is also the risk that the results will not be what you expected. But an experienced plastic surgeon can usually give you a very clear idea of what to expect after surgery.
What To Think About
As with other cosmetic procedures, you are more likely to be happy with the results of your face-lift if you have clear, realistic expectations about what the surgery can achieve and if you share these with your plastic surgeon.
Insurance companies do not cover the costs of face-lifts. It is important to find out what the total costs of the procedure will be, including fees for the operating facility, the anesthesiologist's and surgeon's fees, medicines, office visits, and other services and materials.
Complete the surgery information form (PDF) to help you prepare for this surgery.
By Healthwise Staff Primary Medical Reviewer Anne C. Poinier, MD - Internal Medicine Specialist Medical Reviewer Keith A. Denkler, MD - Plastic Surgery
Current as of March 12, 2017
Empirical Calibration and Minimum-Variance Delta Under Log-Normal Stochastic Volatility Dynamics
Artur Sepp
November 17, 2017
We consider calibration of log-normal stochastic volatility model and computation of option delta consistently with statistical dynamics of the asset price and its implied volatility surface. We introduce the concept of volatility skew-beta which serves as an empirical adjustment for empirical option delta. We show how to calibrate the model and make it consistent with any dynamics of implied volatility under the statistical measure and reproduce empirical option delta. The calibrated model minimizes realized volatility of delta-hedging P&L-s, especially so for non-vanilla options. We present empirical investigation using implied and realized volatilities of four major stock indices (S&P 500, FTSE 100, Nikkei 225, and STOXX 50) to validate the assumption about log-normality of both implied and realized volatilities.
Number of Pages in PDF File: 42
Keywords: Lognormal stochastic volatility, Jumps in price and volatility, Model calibration, Implied volatility skew, Closed-form solution, Option pricing, Minimum-variance hedging
JEL Classification: C00, G00
Date posted: January 30, 2017 ; Last revised: March 9, 2017
Cita Sugerida
Sepp, Artur, Empirical Calibration and Minimum-Variance Delta Under Log-Normal Stochastic Volatility Dynamics (November 17, 2017). Available at SSRN: http://ssrn. com/abstract=2387845 or http://dx. doi. org/10.2139/ssrn.2387845
Asset & Model Coverage
A cutting-edge, production-quality analytics library with a comprehensive set of analytic measures and numerical techniques.
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ISDA standard model
Curve calibration
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Productos básicos
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Modelling data from any external system
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Sensitivities to parameters and market quotes (curves and volatilities)
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Modelos
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Several numerical implementations (analytical formulas, approximation, numerical integration, stress, PDE. Fourier ) available.
The OpenGamma Analytics Library is one of the most flexible systems available for curve calibration. All interest rate models are implemented in the multi-curve framework.
Compatible with OIS discounting and collateral approaches and allows calibration with any instrument (IRS, OIS, futures, FX swaps, X-ccy swaps, etc) as well as calibration of multiple curves simultaneously using multi-dimensional root finder.
Accepts complex curve entanglements and supports interpolated curves, functional curves, and spread curves above another curve.
Users can provide exogenous dates as node points in interpolated curves (e. g. central bank meeting), and produce full Jacobian/transition matrices. Algorithmic differentiation has been implemented for the entire curve-building process to achieve reduced computation time.
Algorithmic Differentiation
Algorithmic differentiation is implemented throughout the library from scratch in its adjoint mode. All sensitivity results are computed through this efficient mechanism (no finite difference / bump and recompute).
Provides derivatives with machine-precision accuracy. In most cases, the theoretical bound is achieved: time for derivative and price < 4 time for price (bound independent of number of derivatives). In the time-consuming case of pricing with calibration, a specific implementation can reduce the ratio to 2. This last implementation has been implemented for amortized swaptions with LMM, reducing computation time by more than 50 wrt bump and recompute.
We support one-factor models for European and American options with barrier features; some support for two-factor models.
Research
The library development is supported by nine full-time quantitative analysts, most of them with investment banking experience at senior level. The library contains standard quantitative finance developments as well as proprietary developments. The proprietary developments are regularly published in international quantitative finance journals and presented at major industry conferences.
Fx Options and Smile Risk
Practical issues in FX options and smile risk FX Options and Smile Risk takes readers through the main technicalities of the FX spot and options markets, helping them develop practical trading skills that will enable them to run an FX options book in the real world. It describes how to build FX volatility surfaces in robust and consistent ways and how to use them in the pricing of vanilla and exotic options. It enables readers to effectively hedge exposures to volatility surface and other risks related to exotic options. It's highly focused on the practical aspects of the pricing and hedging of the typical risks of an FX options desk and deals with the momentous issues of building consistent volatility matrices and a unified approach to pricing and hedging. Antonio Castagna (Milan, Italy) is a Consultant at Iason Ltd, providing pricing and risk management expertise for complex products. He has extensive experience in FX and derivatives, and was previously Head of Volatility Trading at Banca IMI Milan, where he set up the bank's FX Option desk.
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Principles of Financial Engineering, 3rd Edition
Características principales
The Third Edition presents three new chapters on financial engineering in commodity markets, financial engineering applications in hedge fund strategies, correlation swaps, structural models of default, capital structure arbitrage, contingent convertibles and how to incorporate counterparty risk into derivatives pricing, among other topics.
Additions, clarifications, and illustrations throughout the volume show these instruments at work instead of explaining how they should act
The solutions manual enhances the text by presenting additional cases and solutions to exercises
Descripción
Three new chapters, numerous additions to existing chapters, and an expanded collection of questions and exercises make this third edition of Principles of Financial Engineering essential reading. Between defining swaps on its first page and presenting a case study on its last, Robert Kosowski and Salih Neftci's introduction to financial engineering shows readers how to create financial assets in static and dynamic environments. Poised among intuition, actual events, and financial mathematics, this book can be used to solve problems in risk management, taxation, regulation, and above all, pricing.
financial engineers, quantitative analysts in banks and investment houses, and other financial industry professionals; graduate students in financial engineering and financial mathematics programs.
Robert Kosowski
Robert Kosowski is Associate Professor in the Finance Group of Imperial College Business School, Imperial College London, and Director of the Risk Management Lab and Centre for Hedge Fund Research. Robert is an associate member of the Oxford-Man Institute of Quantitative Finance at Oxford University and a member of AIMA's research committee. His research interests include asset management, asset pricing, and financial econometrics with a focus on hedge and mutual funds, performance measurement, asset allocation, business cycles, and derivative trading strategies. Robert's research has been featured in "The Financial Times" and "The Wall Street Journal" and was awarded the European Finance Association 2007 Best Paper Award, an INQUIRE UK 2008 best paper award, an INQUIRE Europe 2009/10 and 2012/13 best paper award, and the British Academy's mid-career fellowship (2011-2012). Robert's research has been published in top peer-reviewed finance journals such as "The Journal of Finance," "The Journal of Financial Economics" and the "Review of Financial Studies." Prior to joining Imperial College London Robert was an Assistant Professor of Finance at INSEAD, where he taught in the MBA, Executive Education, and Ph. D. Programas. Robert was a visiting scholar at the UCSD Economics Department (2000) and the International Monetary Fund (2008). At Imperial Robert teaches in the MSc Finance. He won teaching prizes at Imperial College Business School in 2009 and 2017. Robert holds a BA (First Class Honours) and MA in Economics from Trinity College, Cambridge University, and a MSc in Economics and Ph. D. from the London School of Economics. He has consulted for private and public sector organizations and has worked for Goldman Sachs, the Boston Consulting Group, and Deutsche Bank. His policy related advisory work includes: Specialist Adviser to UK House of Lords (2009-2010) and Expert Technical Consultant (International Monetary Fund, USA, 2008).
Affiliations and Expertise
Associate Professor of Finance and Director of the Risk Management Lab and Centre for Hedge Fund Research, Imperial College, London, UK
Recent Publication
Salih Neftci
Professor Neftci completed his Ph. D. at the University of Minnesota and was head of the FAME Certificate program in Switzerland. He taught at the Graduate School, City University of New York; ICMA Centre, University of Reading; and at the University of Lausanne. He was also a Visiting Professor in the Finance Department at Hong Kong University of Science and Technology. Known his books and articles, he was a regular columnist for CBN daily, the most influential financial newspaper in China.
Affiliations and Expertise
Late of the Global Finance Master’s Program, New School for Social Research, New York, NY, USA
Recent Publication
eBook: EUR 75,95 Formats: PDF, VST (VitalSource Bookshelf), EPUB
Principles of Financial Engineering, 3rd Edition
Dedication
Preface to the Third Edition
Chapter 1. Introduction
1.1 A Unique Instrument
1.2 A Money Market Problem
1.3 A Taxation Example
1.4 Some Caveats for What Is to Follow
1.5 Trading Volatility
1.6 Conclusions
Suggested Reading
Ceremonias
Chapter 2. Institutional Aspects of Derivative Markets
2.1 Introduction
2.2 Markets
2.3 Players
2.4 The Mechanics of Deals
2.5 Market Conventions
2.6 Instruments
2.7 Positions
2.8 The Syndication Process
2.9 Conclusions
Suggested Reading
Ceremonias
Chapter 3. Cash Flow Engineering, Interest Rate Forwards and Futures
3.1 Introduction
3.2 What Is a Synthetic?
3.3 Engineering Simple Interest Rate Derivatives
3.4 LIBOR and Other Benchmarks
3.5 Fixed Income Market Conventions
3.6 A Contractual Equation
3.7 Forward Rate Agreements
3.8 Fixed Income Risk Measures: Duration, Convexity and Value-at-Risk
3.9 Futures: Eurocurrency Contracts
3.10 Real-World Complications
3.11 Forward Rates and Term Structure
3.12 Conventions
3.13 A Digression: Strips
3.14 Conclusions
Suggested Reading
Appendix‗Calculating the Yield Curve
Ceremonias
Chapter 4. Introduction to Interest-Rate Swap Engineering
4.1 The Swap Logic
4.2 Applications
4.3 The Instrument: Swaps
4.4 Types of Swaps
4.5 Engineering Interest-Rate Swaps
4.6 Uses of Swaps
4.7 Mechanics of Swapping New Issues
4.8 Some Conventions
4.9 Additional Terminology
4.10 Conclusions
Suggested Reading
Ceremonias
Chapter 5. Repo Market Strategies in Financial Engineering
5.1 Introduction
5.2 Repo Details
5.3 Types of Repo
5.4 Equity Repos
5.5 Repo Market Strategies
5.6 Synthetics Using Repos
5.7 Differences Between Repo Markets and the Impact of the GFC
5.8 Conclusions
Suggested Reading
Ceremonias
Chapter 6. Cash Flow Engineering in Foreign Exchange Markets
6.1 Introduction
6.2 Currency Forwards
6.3 Synthetics and Pricing
6.4 A Contractual Equation
6.5 Applications
6.6 Conventions for FX Forward and Futures
6.7 Swap Engineering in FX Markets
6.8 Currency Swaps Versus FX Swaps
6.9 Mechanics of Swapping New Issues
6.10 Conclusions
Suggested Reading
Ceremonias
Chapter 7. Cash Flow Engineering and Alternative Classes (Commodities and Hedge Funds)
7.1 Introduction
7.2 Parameters of a Futures Contract
7.3 The Term Structure of Commodity Futures Prices
7.4 Swap Engineering for Commodities
7.5 The Hedge Fund Industry
7.6 Conclusions
Suggested Reading
Ceremonias
Chapter 8. Dynamic Replication Methods and Synthetics Engineering
8.1 Introduction
8.2 An Example
8.3 A Review of Static Replication
8.4 “ Ad Hoc ” Synthetics
8.5 Principles of Dynamic Replication
8.6 Some Important Conditions
8.7 Real-Life Complications
8.8 Conclusions
Suggested Reading
Ceremonias
Chapter 9. Mechanics of Options
9.1 Introduction
9.2 What is an Option?
9.3 Options: Definition and Notation
9.4 Options as Volatility Instruments
9.5 Tools for Options
9.6 The Greeks and Their Uses
9.7 Real-Life Complications
9.8 Conclusion: What is an Option?
Suggested Reading
Appendix 9.1
Appendix 9.2
Ceremonias
Chapter 10. Engineering Convexity Positions
10.1 Introduction
10.2 A Puzzle
10.3 Bond Convexity Trades
10.4 Sources of Convexity
10.5 A Special Instrument: Quantos
10.6 Conclusions
Suggested Reading
Ceremonias
Chapter 11. Options Engineering with Applications
11.1 Introduction
11.2 Option Strategies
11.3 Volatility-Based Strategies
11.4 Exotics
11.5 Quoting Conventions
11.6 Real-World Complications
11.7 Conclusions
Suggested Reading
Ceremonias
Chapter 12. Pricing Tools in Financial Engineering
12.1 Introduction
12.2 Summary of Pricing Approaches
12.3 The Framework
12.4 An Application
12.5 Implications of the Fundamental Theorem
12.6 Arbitrage-Free Dynamics
12.7 Which Pricing Method to Choose?
12.8 Conclusions
Suggested Reading
Appendix 12.1 Simple Economics of the Fundamental Theorem
Ceremonias
Chapter 13. Some Applications of the Fundamental Theorem
13.1 Introduction
13.2 Application 1: The Monte Carlo Approach
13.3 Application 2: Calibration
13.4 Application 3: Quantos
13.5 Conclusions
Suggested Reading
Ceremonias
Chapter 14. Fixed Income Engineering
14.1 Introduction
14.2 A Framework for Swaps
14.3 Term Structure Modeling
14.4 Term Structure Dynamics
14.5 Measure Change Technology
14.6 An Application
14.7 In-Arrears Swaps and Convexity
14.8 Cross-Currency Swaps
14.9 Differential (Quanto) Swaps
14.10 Conclusions
Suggested Reading
Ceremonias
Chapter 15. Tools for Volatility Engineering, Volatility Swaps, and Volatility Trading
15.1 Introduction
15.2 Volatility Positions
15.3 Invariance of Volatility Payoffs
15.4 Pure Volatility Positions
15.5 Variance Swaps
15.6 Real-World Example of Variance Contract
15.7 Volatility and Variance Swaps Before and After the GFC‗The Role of Convexity Adjustments?
15.8 Which Volatility?
15.9 Conclusions
Suggested Reading
Ceremonias
Chapter 16. Correlation as an Asset Class and the Smile
16.1 Introduction to Correlation as an Asset Class
16.2 Volatility as Funding
16.3 Smile
16.4 Dirac Delta Functions
16.5 Application to Option Payoffs
16.6 Breeden‗Litzenberger Simplified
16.7 A Characterization of Option Prices as Gamma Gains
16.8 Introduction to the Smile
16.9 Preliminaries
16.10 A First Look at the Smile
16.11 What Is the Volatility Smile?
16.12 Smile Dynamics
16.13 How to Explain the Smile
16.14 The Relevance of the Smile
16.15 Trading the Smile
16.16 Pricing with a Smile
16.17 Exotic Options and the Smile
16.18 Conclusions
Suggested Reading
Ceremonias
Chapter 17. Caps/Floors and Swaptions with an Application to Mortgages
17.1 Introduction
17.2 The Mortgage Market
17.3 Swaptions
17.4 Pricing Swaptions
17.5 Mortgage-Based Securities
17.6 Caps and Floors
17.7 Conclusions
Suggested Reading
Ceremonias
Chapter 18. Credit Markets: CDS Engineering
18.1 Introduction
18.2 Terminology and Definitions
18.3 Credit Default Swaps
18.4 Real-World Complications
18.5 CDS Analytics
18.6 Default Probability Arithmetic
18.7 Pricing Single-Name CDS
18.8 Comparing CDS to TRS and EDS
18.9 Sovereign CDS
18.10 Conclusions
Suggested Reading
Ceremonias
Chapter 19. Engineering of Equity Instruments and Structural Models of Default
19.1 Introduction
19.2 What Is Equity?
19.3 Equity as the Discounted Value of Future Cash Flows
19.4 Equity as an Option on the Assets of the Firm
19.5 Capital Structure Arbitrage
19.6 Engineering Equity Products
19.7 Conclusions
Suggested Reading
Ceremonias
Chapter 20. Essentials of Structured Product Engineering
20.1 Introduction
20.2 Purposes of Structured Products
20.3 Structured Fixed-Income Products
20.4 Some Prototypes
20.5 Conclusions
Suggested Reading
Ceremonias
Chapter 21. Securitization, ABSs, CDOs, and Credit Structured Products
21.1 Introduction
21.2 Financial Engineering of Securitization
21.3 ABSs Versus CDOs
21.4 A Setup for Credit Indices
21.5 Index Arbitrage
21.6 Tranches: Standard and Bespoke
21.7 Tranche Modeling and Pricing
21.8 The Roll and the Implications
21.9 Regulation, Credit Risk Management, and Tranche Pricing
21.10 New Index Markets
21.11 Structured Credit Products
21.12 Conclusions
Suggested Reading
Ceremonias
Chapter 22. Default Correlation Pricing and Trading
22.1 Introduction
22.2 Two Simple Examples
22.3 Standard Tranche Valuation Model
22.4 Default Correlation and Trading
22.5 Delta Hedging and Correlation Trading
22.6 Real-World Complications
22.7 Default Correlation Case Study: May 2005
22.8 Conclusions
Suggested Reading
Appendix 22.1 Some Basic Statistical Concepts
Ceremonias
Chapter 23. Principal Protection Techniques
23.1 Introduction
23.2 The Classical Case
23.3 The CPPI
23.4 Modeling the CPPI Dynamics
23.5 An Application: CPPI and Equity Tranches
23.6 Differences Between CPDO and CPPI
23.7 A Variant: The DPPI
23.8 Application of CPPI in the Insurance Sector: ICPPI
23.9 Real-World Complications
23.10 Conclusions
Suggested Reading
Ceremonias
Chapter 24. Counterparty Risk, Multiple Curves, CVA, DVA, and FVA
24.1 Introduction
24.2 Counterparty Risk
24.3 Credit Valuation Adjustment
24.4 Debit Valuation Adjustment
24.5 Bilateral Counterparty Risk
24.6 Hedging Counterparty Risk
24.7 Funding Valuation Adjustment
24.8 CVA Desk
24.9 Choice of the Discount Rate and Multiple Curves
24.10 Conclusions
Suggested Reading
Ceremonias
Referencias
Índice
Quotes and reviews
"This text has quickly become a modern classic of financial engineering, as broad in coverage as it is deep in content, and the addition of Kosowski brings another dimension of academic rigor and practical relevance to Neftci's impressive pedagogical legacy." - Andrew W. Lo, MIT Sloan School of Management
"I’m delighted that this classical text has been updated by Professor Kosowski to reflect financial engineering post-crisis. This timely combination of timeless principles and recent revelations makes for an irresistible read." --Peter Carr, Morgan Stanley and New York University
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When was the last time you entered a trade and it immediately moved against you even though you felt confident the market was going to move in your favor? When was the last time you traded a breakout and got stopped out? I’m willing to bet you’ve experienced one or both of these things recently in your own trading, and I’m also willing to bet that me or one of my students probably took the opposite side of one of these trades that seemed to ‘fake you out’ of your position…
You see, false-breaks happen all the time in the markets; they are a result of the ‘herd mentality’ that causes people to buy the top of a move or sell the bottom. As price action traders, we are in a unique position to take advantage of false-breaks and of the weak ‘herd mentality’ that so many amateur traders possess.
I have made most of my money as a trader by using contrarian trading approaches like false-breaks and my proprietary fakey trading strategy. It is the power of contrarian trading and using false-break patterns and fakey setups that allows myself and other savvy price action traders to profit from other traders’ misfortunes. This may sound a little harsh, but it’s the reality of trading that the majority of traders lose money, informed and skilled traders make money, and the ‘pigs get slaughtered’, as the saying goes. I hope there are light bulbs going off in your head now . because this article is all about contrarian thinking, false-breaks, and how to take advantage of the ‘herd mentality’ that causes so many traders to enter right when the market is about to change direction…
So what exactly is a false-break?
I thought you’d never ask! Joking, I know you are probably thinking that right now, so here you go…
A false-break can be defined as a ‘deception’ by the market; a test of a level that results in a break of that level but the market then retracts and does not sustain itself above or below that level. In other words, the market does not close outside of the level being tested; rather it leaves behind a false-break of it. These false-breaks are huge pieces of evidence for impending market direction, and we need to learn to use them to our advantage instead of becoming their victim.
Here is a visual example of a false-break of a key market level:
Essentially, a false-break can be thought of as a contrarian move that ‘sucks’ the over-committed side of the market out. The concept is to wait for the price movement to clearly show that a market has committed to one side of a trade and that they would be ‘forced’ to liquidate their position(s) on a strong reversal in the other direction. Typically, we see these scenarios unfold as a trending market becomes extended and all the amateurs jump in right before the counter-trend retrace, or at key support and resistance levels or at consolidation breakout scenarios.
The herd mentality causes traders to enter the market typically only when it ‘feels’ safe. However, this is the deception; trading off feeling and emotion is exactly why most traders lose money in the markets. Many traders become deceived because the market looks very strong or very weak, so they think it’s a no-brainer to just jump in with that momentum. However, the truth of the matter is that markets ebb and flow and they never move in a straight line for very long. This is known as “reversion to the mean” and it’s something I expand on significantly in my advanced Forex trading course .
We really have to use logic and counter-intuitive or ‘contrarian’ thinking to profit off of the weak-minded herd mentality that dominates most traders’ minds. This is why it’s very important to remain disciplined in the area of trading false-breaks, rejections and failures, and why I love trading them so much.
Types of False Breaks
1. Classic Bull and Bear traps at key market levels
A bull or bar trap is typically a 1 to 4 bar pattern that is defined by a false-break of a key market level. These false-breaks occur after large directional moves and as a market approaches a key level. Most traders tend to think a level will break just because a market has approached it aggressively, they then buy or sell the breakout and then many times the market will ‘fake them out’ and form a bull or bear trap.
A bull trap forms after a move higher, the amateurs who were on the sidelines watching a recent strong move unfold cannot take the temptation anymore, and they jump in just above or at a key resistance level since they feel confident the market now has the momentum to break above it. The market then breaks slightly above the level and fills all breakout orders, and then falls lower as the big boys come in and push the market lower, leaving the amateurs ‘trapped’ in a losing long position.
2. False-break of consolidation
False breaks of consolidation or trading ranges are very common. It’s easy to fall into the trap of thinking a trading range is going to breakout, only to see it reverse back into the body of the range. The best way to avoid this trap is to simply wait until there is a clear close outside of the trading range on the daily chart, and then you can begin to look for price action trading signals in the direction of the breakout.
3. Fakey’s (inside bar false-breaks)
The Fakey setup is one of my all-time favorite price action setups and learning to trade it will do a lot for helping you to understand market dynamics. Essentially, the Fakey is a price action pattern that requires there to be a false-break of an inside bar setup. So, once you have an inside bar setup, you can watch for a false-break of the inside bar and the mother bar. Now, I am not going to get into all the different versions of the fakey trading strategy today or the different ways to trade it, but you can learn everything about my proprietary forex fakey trading strategy in my professional Forex trading course.
Here’s an image of two Fakey setups, note that one has a pin bar as the false-break and other does not, these are just two of the variations of the Fakey setup:
False-breaks can create long-term trend changes
As price action traders. we can use the price action of a market to anticipate false-breaks and look for them at key levels as they will often set off significant changes in price direction or even a change in trend from these key levels.
We need to pay attention to the ‘tails’ of candles that occur at or near key levels in the market. Ask yourself how prices reacted during each daily session…where did they close? The close is the most important level of the day, and often if a market fails to close beyond a key market level, it can signal a significant false-break. Often, prices will probe a level or attempt to break out, but by the close of the daily bar price has rejected that level and ‘tailed out’, showing a false-break or false-test of the level. A failure of the market to close beyond a key market level can lead to a large retracement or a change of trend. Thus, the close of a price bar is the most important level to watch, and the daily chart close is what I consider to be the most important.
Here’s an example of a false-break in the EURUSD daily chart that led to a top in the market and started a long-term downtrend:
History Teaches Us A Lesson
It’s worth noting that on the week famous trader George Soros shorted the British pound and ‘broke’ the Bank of England ( September 16, 1992) – the chart had shown a massive false-break signal. The chart below shows the price breaking upwards to new highs and then crashing back down. To those who follow me regularly you will note that this was actually a classic fakey setup, and is clear evidence that this price action strategy has worked for decades.
Final word on false-breaks…
As traders, if we don’t learn to anticipate and identify deceptions or ‘false-breaks’ in the market, we will lose money to traders who do. If we pay attention to the price action at key levels on the daily chart time frame, the ‘writing’ is usually on the wall in regards to false-breaks.
If I had to leave you with one crucial piece of advice for your Forex trading career, it would be to drop everything right now and start studying false-breaks and contrarian trading approaches. By doing so, you will be ahead of 95% of traders who are stuck in a cycle of trading off mainstream misconceptions and ineffective trading methods. As a contrarian, I want to be trading when most other retail traders are committed to the wrong side of the market, and this is difficult to do if you don’t understand false-breaks and fakey patterns. Trading false-breaks and my proprietary ‘fakey setup’ is a core focus in my Forex price action trading course. and I expand on these topics in great detail in it. I teach my students a plethora of different price patterns to look out for when trading false-breaks and fakey setups. This ‘contrarian’ style of trading is something I strongly believe in, and it has proven itself time and time again. If you were to learn only one single trading strategy to apply in your Forex trading, false-breaks would be on top of the list.
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Robustly hedging variable annuities with guarantees under jump and volatility risks.
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MLA style: "Robustly hedging variable annuities with guarantees under jump and volatility risks.." The Free Library . 2007 American Risk and Insurance Association, Inc. 17 Mar. 2017 http://www. thefreelibrary. com/Robustly+hedging+variable+annuities+with+guarantees+under+jump+and. - a0164636749
Estilo de Chicago: La biblioteca libre. S. v. Robustly hedging variable annuities with guarantees under jump and volatility risks.." Retrieved Mar 17 2017 from http://www. thefreelibrary. com/Robustly+hedging+variable+annuities+with+guarantees+under+jump+and. - a0164636749
APA style: Robustly hedging variable annuities with guarantees under jump and volatility risks. (n. d.) >The Free Library. (2017). Retrieved Mar 17 2017 from http://www. thefreelibrary. com/Robustly+hedging+variable+annuities+with+guarantees+under+jump+and. - a0164636749
Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black-Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black-Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at-the-money implied volatility is a state variable. We compute risk minimization hedging by modeling at-the-money Black-Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.
Until the early 1990s, most variable annuity contracts provided a modest minimum death benefit guarantee of return of premiums. This is the greater of the account balance at death and the sum of the premium deposits, less partial withdrawals, since inception of the contract. The cost of this benefit was considered insignificant and insurance companies did not assess an explicit charge and did not hold an additional reserve for this benefit.
Since the middle of 1990s, variable annuity contracts have offered more substantial minimum death benefit guarantees (GMDB). These new guarantees typically include one or more of the following features (Hill, 2003; Longley-Cook and Kehsberg, 2003; Scheinerman, Kleiman, and Andrews, 2001):
* Reset: The benefit is the periodically automatically adjusted account balance plus the sum of premium deposits less withdrawals since the last reset date.
* Roll-up: The death benefit is the larger of the account balance on the date of the death and the accumulation of premium deposits less partial withdrawals accumulated at a specified interest rate. There may be an upper bound on the benefits.
* Ratchet: The benefit is similar to a reset benefit except that it is now redetermined at the end of a pre set number of years (typically annually) to be the larger of the current account balance and the account balance on the prior ratchet date. In contrast to a reset benefit, the ratchet benefit is not allowed to decrease.
Typically variable annuities have combinations of different features. In addition to these new benefits, more aggressive equity-indexed annuities (EIA) have been the fastest growing annuity products since 1995. The EIA is an annuity in which the policyholder's rate of return is determined as a defined share in the appreciation of an outside index, e. g. S&P 500 in the United States, with a guaranteed minimum return.
While these new insurance contracts provide policyholders with downside risk protection, insurers are faced with the challenging task of hedging against the unfavorable movement of the equity values. Recent research has applied no arbitrage pricing theory from finance to calculate the values of the embedded options in insurance contracts; see, for example, Brennan and Schwartz (1976), Boyle and Schwartz (1977), Aase and Persson (1994), Boyle and Hardy (1997), Bacinello and Persson (2002), Milesky and Posner (2001), Pelsser (2003), Pennacchi (1999), and Persson and Aase (1997).
Although much of the emphasis of the literature has been on pricing, hedging the embedded options in these new insurance policies is of crucial importance for risk management. Hedging options embedded in the variable annuities is particularly difficult since maturities of insurance contracts are long (typically longer than 10 years), transaction costs limit the rebalancing frequency of hedging portfolios, and liquidity restricts the choice of possible hedging instruments. In addition, due to sensitivity of the benefits to the tail distributions, a simple Black-Scholes model for the underlying equity price is not adequate for the observed fat tails of the equity return distributions; a model incorporating jump risk and/or volatility risk is necessary. The long maturity of insurance contracts also makes interest risk modeling necessary; we discuss hedging strategy computation under the interest risk in a separate article (Coleman, Li, and Patron, 2006). Finally, there are additional risks such as basis risk, mortality risk, and surrender risk; these risks are not addressed here.
In this article, we focus on computing and evaluating hedging effectiveness of strategies using either the underlying (futures) or standard options as hedging instruments, in order to control the market risk embedded in the insurance contracts under different models, including models suitable for fat tails of return distributions. We note that this is different from the emphasis of the current literature which focuses on the fair value computation of the insurance contracts. However, hedging strategy computation does automatically provide an estimate of the hedging cost. To focus on modeling and the hedging strategy computation, we assume in this article that the underlying account is linked directly to a market index, e. g. S&P 500; the issue of basis risk is not addressed here.
For a derivative contract, the most frequently used hedging strategy in the financial industry is delta hedging. In a delta hedging strategy, the trading position of the underlying is computed from the sensitivity (first order derivative) of a (risk adjusted) option value to the underlying. Hence the delta hedging strategy is determined from the underlying price dynamics under a risk adjusted measure, which is unique under a Black-Scholes model but not under a jump diffusion model.
Unfortunately, delta hedging is only instantaneous and continuous rebalancing is clearly impossible in practice. Under the assumption that a hedging portfolio can only be rebalanced at discrete times, the market is incomplete and the intrinsic risk of an option cannot be completely eliminated. For hedging effectiveness and risk management analysis, one is interested in the real world performance: a hedging strategy computed under a risk adjusted measure may not be optimal under the real world price dynamics when the market is incomplete due to discrete hedging, insufficient hedging instruments, jump risk, and volatility risk.
Given that it is practically impossible to completely eliminate the risk in an option, can we determine a hedging strategy to minimize a chosen measure of risk under a real world price dynamics? A risk minimization hedging method computes an optimal dynamic trading strategy to minimize a specific measure of risk under a real world price model (e. g. Follmer and Schweizer, 1989; Schal, 1994; Schweizer, 1995, 2001; Mercurio and Vorst, 1996; Heath, Platen, and Schweizer 2000a, b). Given a statistical model and assuming trading is done at a discrete set of times, a local risk minimization hedging strategy is computed at each trading time to minimize the variance of the difference between the liquidating portfolio value and the value of the portfolio for the next hedging period.
In order for a discrete hedging strategy using the underlying to be effective, hedging portfolios need to be rebalanced frequently. This can be problematic for hedging the embedded option in an insurance contract due to the transaction costs and long maturity of insurance contracts. In addition, for variable annuities with guarantees, the long maturity and sensitivity of the benefits to the tails of the account value distribution make the choice of a suitable model difficult but crucial. For example, in order to accurately model the tails of the underlying distribution, jump risk and volatility risk may need to be considered. The presence of such additional risks deteriorates the effectiveness of hedging using the underlying. Since option markets have become increasingly more liquid, standard options are often used as hedging instruments for a complex option. In this article, we also compare discrete risk minimization hedging using the underlying with that of using liquid standard options.
Using standard options as hedging instruments adds additional complexity in equity return modeling; in this situation it is imperative that stochastic implied volatilities be adequately modeled. When quantifying and minimizing risk, a statistical model for changes of the underlying and changes of hedging instrument prices between each rebalancing time is needed. In particular, if standard options are used as hedging instruments, it is important to adequately model the stochastic implied volatilities. Unfortunately, estimating a model which is capable of prescribing evolution of implied volatilities for a long time horizon is a challenging task. There typically are two possible approaches for model estimation. One approach is based on historical prices and the other is based on market calibration. Historical model estimation is faced with a complex decision of the choice of data and statistical methods which lead to stable estimation. Market calibration starts with calibrating a model for the underlying risk from the current liquid option prices. Theoretical values of the option to be hedged (as well as standard options if used as hedging instruments) are computed based on the calibrated model and then either a delta hedging strategy or option hedging strategy is determined based on this risk adjusted valuation. A difficulty of this calibration approach is that the current liquid options have short maturities and the model calibrated from current prices is unlikely to be able to prescribe the option price dynamics for the long time horizon of the insurance contracts; thus, the hedging strategy determined based on market calibration is exposed to significant model risks, particularly when the hedging portfolio needs to be rebalanced frequently. In addition, when the underlying price model is calibrated to the current market option prices, the calibrated model describes price dynamics under a risk adjusted measure, not the objective probability measure. In an incomplete market, it is necessary to quantify and evaluate hedging effectiveness under the real world price dynamics. A hedging strategy which is determined based on a risk adjusted valuation is typically not optimal when evaluated under the real world price dynamics; the more difficult the option is to hedge, the less optimal will be the hedging strategy determined under a risk adjusted measure.
When standard options are used as hedging instruments, it is crucial that stochastic implied volatilities (which yield the hedging instrument prices) are suitably modeled. This is difficult to accomplish by calibrating a model for the underlying and determining the option prices based on the calibrated model. The increasing autonomy of the option market also leads to a question of whether it is possible to accurately model market standard option evolution in this fashion. We propose to compute the risk minimization hedging using standard options by jointly modeling the underlying price dynamics and the Black-Scholes at-the-money implied volatility explicitly. This modeling approach has many advantages. Firstly, implied volatilities are readily observable and hedging positions can be adjusted according to these observable variables. Secondly, the standard option hedging instruments can be easily and accurately priced. Thirdly, calibration to the current option market is automatically done by setting the implied volatilities to the market implied volatilities. Finally, the statistical information of the past implied volatility evolution can be incorporated in the model.
The main contribution of this article is hedging effectiveness evaluation for embedded options in variable annuity guarantees, under models for jump and volatility risks. We compare delta hedging, risk minimization hedging using the underlying, and risk minimization hedging using standard options. We illustrate that the risk minimization discrete hedging using the underlying can be significantly more effective than popular delta hedging, particularly when rebalancing is infrequent. We show that, under a jump risk, risk minimization hedging using standard options is more effective in risk reduction than discrete hedging using the underlying. We propose to model the implied volatilities directly when computing a hedging strategy using standard options as hedging instruments. We evaluate hedging effectiveness under a joint dynamics of the underlying and at-the-money implied volatility and illustrate that risk minimization hedging using standard options can be more effective than using the underlying under both jump and volatility risks.
Due to a potentially excessive liability, hedging the lookback option embedded in a ratchet GMDB is a more difficult problem than hedging options embedded in a GMDB with return of premium, roll-up or reset guarantees. Hence we focus on hedging risk induced from the ratchet feature. Though inclusion of additional benefit features would increase the computational complexity of the problem, we do not believe that it affects the methodology and main conclusions regarding risk minimization hedging under jump and volatility risks.
Two of the main types of risk affecting the VA with ratchet GMDB are the mortality risk and the market risk. By the Law of Large Numbers, if the number of insurance policies is large enough, the loss due to mortality is approximately equal to the expected loss. In this article, we focus on the market risk and thus assume that the mortality risk can be diversified away. We analyze, in particular, the performance of hedging strategies under jump and volatility risks. We first describe, in the section "Hedging GMDB in Variable Annuities," computation of dynamic discrete risk minimization hedging strategies, using either the underlying or standard options, for the options embedded in the GMDB with a ratchet feature. We show that the risk minimization hedging using the underlying as the hedging instrument outperforms the delta hedging strategy in the section "Hedging Using the Underlying Asset." We then compare the hedging effectiveness of using the underlying with that of using liquid options in the section "Hedging Using Standard Options" and we illustrate that, in the absence of volatility risks, risk minimization hedging using liquid option instruments can be significantly more effective than hedging using the underlying. Finally, in the section "Hedging Under Volatility Risks," we compute a risk minimization hedging when at-the-money implied volatility follows an Ornstein-Uhlenbeck process and compare its hedging effectiveness with that of using the underlying under both instantaneous and implied volatility risk. We illustrate that, when implied volatility risk is suitably modeled, risk minimization hedging using options can potentially be more robust and provide better risk reduction than hedging using the underlying.
HEDGING GMDB IN VARIABLE ANNUITIES
For risk management of variable annuities, one is interested in risk quantification and risk reduction under a statistical model for the real price dynamics. In this article, we are interested in computing an optimal hedging strategy, under an assumed statistical market price model, for a variable annuity which has a minimum death guarantee benefit with a ratchet feature. We first describe a delta hedging computation, based on the risk adjusted fair value computation, and risk minimization hedging based on an incomplete market assumption.
Assume that the benefit depends on the value of an equity linked insurance account with anniversaries at
0 = [f. sub.0] < [f. sub.1] <. < [f. sub.[??]-1] < [f. sub.[??]] = T.
Assume that the account value at time [f. sub. i] is. Since we assume that the mortality risk can be diversified, we consider here the hedging problem with a fixed maturity T. The option embedded in a variable annuity with a ratchet GMDB is a path dependent lookback option. The payoff of GMDB with a ratchet feature is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] denotes the running max account value up to the anniversary time [f. sub. i-1], i. e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
Note that [f. sub.[??]] = T. Let [[PI].sub. T] denote the second term in the payoff in (1), i. e. [[PI].sub. T] = max ([H. sub. T] - [S. sub. T], 0) (recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]). The benefit of GMDB with a ratchet feature equals the account value plus a lookback put option with payoff [[PI].sub. T].
Most hedging strategies used in the financial industry are computed based on a risk adjusted option valuation. In particular, at any time t and underlying price St, a delta hedging strategy is determined by first computing the option value V([S. sub. t], t) under a risk adjusted measure and the hedging position in the underlying is given by [differential]V([S. sub. t], t)/[differential]S.
The fair value of a European path dependent option can be computed given a model for the underlying price dynamics under a risk adjusted measure. Consider a European path dependent option whose single payoff at the fixed time T = [f. sub.[??]] depends on the price path of a single underlying asset [S. sub. t]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
Assume more generally that the option payoff at the maturity T is given by
where T = [T. sub.[??]] and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. For a lookback option, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [phi]([H. sub. T], [S. sub. T]) = max([H. sub. T] - [S. sub. T], 0).
The time [f. sub. i] value of such a path dependent option is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
where [E. sup. Q](*) denotes the expectation under a risk adjusted measure and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] denotes information filtration up to [f. sub. i]. The value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] can be computed recursively as follows: given [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for j = M - 1. -1. 0,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
Assume that the transitional density function of [S. sub. t] from [f. sub. j] to [f. sub. j+1], under a risk adjusted measure, is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Entonces
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2)
Note that the only random component of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] comes from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
In practice a hedging portfolio cannot be rebalanced continuously; the rebalancing frequency may be further limited for hedging variable annuities due to the unusually long maturities. Given that a hedging portfolio can be rebalanced only at a limited discrete set of times, what is the optimal hedging strategy? How can this strategy be determined? This is a question of option hedging in an incomplete market and the optimal hedging strategy can be computed to minimize a measure of the hedge risk. In most of the present literature for pricing and hedging in an incomplete market, a hedging strategy is computed to minimize either the quadratic local risk or the quadratic total risk (e. g. Follmer and Schweizer, 1989; Schal, 1994; Schweizer, 1995, 2001; Mercurio and Vorst, 1996; Heath, Platen, and Schweizer 2000a, b). As an alternative to quadratic risk measure, a piecewise linear measure is used in Coleman (in press), Coleman, Li, and Patron (2003), and Patron (2003) to compute discrete local and total risk minimization hedging strategies. The total risk minimization hedging strategy is computed by solving a stochastic optimization problem in which the risk of failure to match the payoff through the dynamic hedging strategy is minimized (see e. g. Follmer and Schweizer, 1989; Schal, 1994; Schweizer, 1995, 2001; Mercurio and Vorst, 1996; Heath, Platen, and Schweizer 2000a, b). F611mer and Schweizer (1989) first consider the local risk measure. A mean variance total risk minimizing strategy is first considered by Schweizer (1992) and Duffie and Richardson (1991). The quadratic criteria for risk minimization in the framework of discrete hedging have been studied in Follmer and Schweizer (1989), Schal (1994), Schweizer (1995), Mercurio and Vorst (1996), and Bertsimas, Kogan, and Lo (2001).
Assume that T > 0 is a hedging time horizon and assume that there are M trading opportunities (it is assumed that, for simplicity, the anniversaries of the variable annuity form a subset of rebalancing times) at
0 = [t. sub.0] < [t. sub.1] <. < [t. sub. M-1] < [t. sub. M] = T.
Suppose that we want to hedge a European option whose payoff at the maturity T is denoted by [[PI].sub. T]. Suppose also that the financial market is modeled by a probability space ([OMEGA], F, P), with filtration [(F. sub. k).sub. k=0, 1. M], and the discounted underlying asset price follows a square-integrable process. As discussed before, we compute optimal hedging strategies under a statistical model for the real world price dynamics. This is essential because we want to quantify and reduce the risk of our hedging strategies and this should be done under the real probability measure. Denote by [P. sub. k] the value of the hedging portfolio at time [t. sub. k] and by [C. sub. k] the cumulative cost of the hedging strategy up to time [t. sub. k]; this includes the initial cost for setting up the hedging portfolio and the additional costs for rebalancing it at the hedging times [t. sub.0]. [t. sub. k].
When trading is limited to a set of discrete times using a finite number of instruments, it is impossible to eliminate the intrinsic option risk. Based on the risk minimization principle, there are two main quadratic approaches for choosing an optimal hedging strategy. One possibility is to control the total risk by minimizing E([([[PI].sub. T] - [P. sub. M]).sup.2]), where E(*) denotes the expected value with respect to the objective probability measure P and [P. sub. M] is portfolio value at [t. sub. M] associated with a self financed trading strategy; thus [P. sub. M] is the initial portfolio value [P. sub.0] plus the cumulative gain. This is the total risk minimization criterion. A total risk minimizing strategy exists under the additional assumption that the discounted underlying asset price has a bounded mean-variance trade-off. In this case, the strategy is given by an analytic formula. The existence and the uniqueness of a total risk minimizing strategy have been extensively studied in Schweizer (1995).
Computing an optimal total risk minimization requires solving a dynamic stochastic programming problem which is, in general, computationally very difficult. In addition, a total risk minimization strategy is weakly dynamically consistent (Mercurio and Vorst, 1996). Local risk minimization hedging, on the other hand, assumes that the hedging portfolio value [P. sub. M] equals the liability [[PI].sub. T] and computes the hedging strategy to minimize each incremental cost E([([C. sub. k+1] - [C. sub. k]).sup.2] | [F. sub. k]) for k = M - 1, M - 2. 0; here [C. sub. k] denotes the cumulative cost of the hedging strategy. Compared to the total risk minimization hedging strategy, computation of a local risk minimization strategy is simpler. In addition, a local risk minimization hedging strategy is strongly dynamically consistent (Mercurio and Vorst, 1996). The same assumption that the discounted underlying asset price has a bounded mean-variance trade-off is sufficient for the existence of an explicit local risk minimizing strategy (see Schal, 1994). This strategy is no longer self-financing, but it is mean-self-financing; i. e. the cumulative cost process is a martingale. In general, the initial costs for the local risk minimizing and total risk minimizing strategies are different. However, Schal (1994) noticed the initial costs agree in the case when the discounted underlying asset price has a deterministic mean-variance trade-off.
In this article, we focus on the local risk minimization hedging. In the current literature on pricing and hedging in an incomplete market, the hedging instruments are typically the underlying assets. In this article, we also consider standard options as hedging instruments; thus, the traded instruments may exist for a subperiod of the entire hedging horizon.
Assume that at time [t. sub. k], n risky hedging instruments with values [U. sub. k] [member of] [R. sup. n] can be traded; we assume that [U. sub. k] are normalized by riskless bonds. At time [t. sub. k+1], these n risky hedging instruments have values [U. sub. k]([t. sub. k+1]); we have omitted dependence on the underlying value for notational simplicity.
A hedging strategy is a sequence of the trading positions in the risky hedging instruments [U. sub. k] and riskless bond, respectively. The hedging position ([[xi].sub. k], [[eta].sub. k]) at time [t. sub. k] is liquidated at time [t. sub. k+1], at which time a new hedging position ([[xi].sub. k+1], [[eta].sub. k+1]) is formed.
The initial cost of the trading portfolio is
[C. sub.0] = [P. sub.0] = [U. sub.0] x [[xi].sub.0] + [[eta].sub.0],
where [U. sub.0] is a n-row vector of the initial hedging instrument values. Similarly, the value of the trading portfolio P at any time [t. sub. k] is
[P. sub. k] = [U. sub. k] x [[xi].sub. k] + [[eta].sub. k].
The cumulative gain of the trading strategy at time [t. sub. k] is
[G. sub. k] = [k-1.summation over j=0]([U. sub. j]([t. sub. j+1]) - [U. sub. j]) x [[xi].sub. j].
Recall that [U. sub. k] denotes the values of the risky assets normalized by riskless bonds and [U. sub. k]([t. sub. k+1]) denotes the normalized values of the same risky assets at [t. sub. k+1]. With this standard notation, the bond value is identically one and does not appear in the cumulative gain.
The cumulative cost of the trading strategy at [t. sub. k] is
[C. sub. k] = [P. sub. k] - [G. sub. k].
A trading strategy is self-financed if
[C. sub. k+1] - [C. sub. k] = [U. sub. k+1] x [[xi].sub. k+1] - [U. sub. k]([t. sub. k+1]) x [[xi].sub.1] + [[eta].sub. k+1] - [[eta].sub. k] = 0, k = 0, 1. M - 1.
When a market is incomplete, a risk minimization hedging strategy is fundamentally different from hedging strategies based on risk adjusted option values, e. g. delta hedging and semi-static hedging which is based on the principle of replicating a complex option with standard options (Carr, 2002; Carr and Wu, 2002). Consider hedging a lookback option using the underlying as an example. Delta hedging, which is based on continuously rebalancing, is computed from the sensitivity of risk adjusted option values to the underlying. Thus delta hedging requires a price dynamics under a risk adjusted measure which is typically obtained by calibration to the standard option market. For hedging lookback options embedded in insurance benefits, there are potentially serious difficulties due to the requirement of market calibration to the option market. Firstly, for risk management and hedging purposes, real world price dynamics models are needed. Such models are essential to accurately quantify risks, for example, in emerging cost analysis and calculating reserves. They are equally essential for hedging strategy computation and analysis. A model for the market price evolution (in contrast to a risk adjusted price dynamics) is needed, particularly when hedging is difficult due to market incompleteness. Secondly, any parsimonious model inevitably leads to calibration errors. Thirdly, if standard options are used as hedging instruments, it is crucial for a model to calibrate to the forward implied volatilities (due to the long maturity). Unfortunately, this is difficult to accomplish. For risk minimization hedging, one typically assumes statistical models for the market value of the underlying and the market values of the hedging instruments at trading times; risk adjusted liability values are not needed. It may be possible to estimate such a model from the historical observable prices of the underlying and hedging instruments. We further discuss this possibility of computing risk minimization hedging strategies using standard options by explicitly modeling implied volatilities in the section "Hedging Under Volatility Risks." Given a statistical model for both the underlying asset and hedging instruments, a risk minimization hedging strategy is computed to minimize hedge risk. Finally, a risk minimization hedging achieves optimality under a chosen risk measure. For example, a quadratic local risk minimization hedging strategy is optimal with respect to the specified trading requirement and the quadratic increment cost. Delta hedging and semi-static hedging are typically not optimal with respect to the discrete trading specification.
HEDGING USING THE UNDERLYING ASSET
In practice, hedging options embedded in the variable annuities is a risk management problem in an incomplete market. In our discussion so far, we have emphasized the theoretical difference between a risk minimization hedging strategy based on an incomplete market assumption and a hedging strategy based on risk adjusted valuation. In this section, we evaluate, computationally, the effectiveness of delta hedging and risk minimization hedging using the underlying asset for a GMDB with a ratchet feature. As discussed in the section "Hedging GMDB in Variable Annuities," the payoff of the option embedded in the variable annuity with a ratchet benefit can be expressed as [[PI].sub. T] = max([H. sub. T] - [S. sub. t], 0) where the path dependent (ratchet) value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
We evaluate hedging performance in terms of the total risk and total cost at the maturity T for the entire hedging horizon from t = 0 to T. Let [P. sup. sf. sub. M] denote the time T self-financed hedging portfolio value corresponding to a hedging strategy. For example, if represents the optimal holdings computed from the risk minimization hedging computation, then
[P. sup. sf. sub. M] = [P. sub.0] + [G. sub. M] = [U. sub.0] x [[xi].sub.0] + [[eta].sub.0] + [M-1.summation over j=0] ([U. sub. j]([t. sub. j+1]) - [U. sub. j]) x [U. sub. j] x [[xi].sub. j].
The total risk [[PI].sub. T] - [P. sup. sf. sub. M] measures the amount of money that the hedging strategy is short of meeting the liability at maturity T. The total cost [[PI].sub. T] - [G. sub. M] is the time T lookback option payoff plus the cumulative loss, -[G. sub. M], of the hedging strategy. Unless stated otherwise, a maturity of 10 years and a zero dividend yield are assumed in the computational results.
We compare the expected total cost and the first two moments of the total risk. Information about liability (payoff) at maturity T is listed under the column [[PI].sub. T] for comparison; it corresponds to no hedging. Results obtained using the underlying for biweekly hedging, monthly hedging, and annually hedging are given under the column biweekly, monthly, and annually, respectively.
In addition, for a given confidence level, we report the value-at-risk (VaR) and the conditional value-at-risk (CVaR) of the total risk at the maturity T. For example, VaR(95 percent) is the minimum amount of money, that the self-financed hedging portfolio [P. sup. sf. sub. M] is short of meeting the liability [[PI].sub. T], with 5 percent probability; it is the cost at T, for the hedger, that is exceeded with 5 percent probability. The value CVaR(95 percent) is the expected cost conditional on the cost exceeding VaR(95 percent).
Although a Black-Scholes model is not adequate for modeling a return distribution with a fat tail, we first assume a Black-Scholes model and consider using the underlying as the hedging instrument; this is interesting since the market incompleteness in this case comes entirely from our inability to hedge continuously. Under a Black-Scholes model, the real world underlying price [S. sub. t] is modeled as a geometric Brownian motion
d[S. sub. t]/[S. sub. t] = [mu] x dt + [sigma] x d[W. sub. t], (3)
where [mu] is the expected rate of asset return, [sigma] is the volatility, and [W. sub. t] is a standard Brownian motion. Assume that the instantaneous risk free rate is r > 0. Note that positions in a delta hedging strategy are independent of the actual expected rate of return [mu].
In our computational results, it is assumed that the initial account value [S. sub.0] = 100. Table 1 compares hedging performance of delta hedging with that of the risk minimization hedging using the underlying. Table 1 illustrates that, under a Black-Scholes model:
* The effectiveness of both risk minimization hedging using the underlying and delta hedging improves significantly as the portfolio is rebalanced more frequently. In particular, the extreme risk of biweekly rebalancing, measured in VaR and CVaR, is noticeably smaller than that of the monthly rebalancing. Naturally, in practice, one needs to evaluate hedging decisions with additional consideration of transaction cost, and possibly consider move-based rather than time-based discrete hedging. It will also be interesting to investigate possibility of including transaction cost consideration explicitly in the risk minimization hedging computation.
* For risk minimization hedging, the initial cost Co decreases as the rebalancing frequency decreases. The initial cost of delta hedging, on the other hand, does not change with the frequency of rebalancing; it equals the unique initial lookback option price. The initial hedging cost of the risk minimization hedging is smaller compared to that of the delta hedging.
* While the delta hedge overhedges (negative mean), the mean of the hedge error for the risk minimization is close to zero. This is due to the fact that delta-hedging does not take hedging frequency into consideration and the holdings are computed in a risk neutral framework, while the hedging performance in Table 1 is evaluated under the "real-world" dynamics. In contrast, risk minimization holdings are computed under the "real-world" dynamics, for a specific hedging frequency (annual, monthly, respectively, biweekly). This explains the average performance difference for the two hedging strategies.
* Compared to delta hedging, risk minimization hedging using the underlying is significantly more effective in reducing risk, measured either in standard deviation, VaR, or CVaR.
* For both delta hedging and risk minimization hedging, the expected total cost increases as the hedging portfolio is rebalanced more frequently. For annual rebalancing, the average total cost of the risk minimization hedging is larger than that of delta hedging. For monthly and biweekly rebalancing, the average total costs of risk minimization are close to that of delta hedging; however the risk from the risk minimization hedging is significantly smaller than that from delta hedging.
Table 1 clearly indicates that, under a Black-Scholes model, risk minimization hedging using the underlying is better than delta hedging since the former minimizes risk under the assumption that hedging portfolio is rebalanced at the specified times.
Unfortunately, a Black-Scholes model (3) is not appropriate for the risk management of variable annuities since the embedded options are sensitive to tails of the underlying distribution; we consider the Black-Scholes model here to illustrate the difference between risk minimization hedging using the underlying and delta hedging when the incompleteness comes from discrete rebalancing, not additional risks such as jump or volatility. Determining a suitable model for the underlying is usually a challenging task; it is even more difficult for variable annuities due to the long maturities and sensitivities of the payoff to the extreme price movements. The Canadian Insurance Association on Segregated Funds (SFTF) recommends that model calibration be adjusted to give a sufficiently accurate fit in the left tail of the distribution (Hardy, 2003). A Black-Scholes model certainly seems a poor choice in this regard. Next we consider Merton's jump diffusion model which is more suitable to model fat tails of return distributions.
Let us assume that the real world price dynamics is modeled by a jump diffusion model with a constant volatility, i. e.
d[S. sub. t]/[S. sub. t] = ([mu] - q - [kappa][lambda]) x dt + [sigma] d[W. sub. t] + (J - 1) x d[[pi].sub. t], (4)
where [mu] is the expected rate of return, q is the continuous dividend yield, and is a constant volatility. In addition, [[pi].sub. t] is a Poisson counting process, [lambda] > 0 is the jump intensity, and J is a random variable of jump amplitude with [kappa] =E(J - 1). For simplicity, log J is assumed here to be normally distributed with a constant mean [[mu].sub. J] and variance [[sigma].sup.2.sub. J] thus E(J) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
While a risk minimization hedging strategy is computed directly from the original process (4), delta hedging is computed from a risk adjusted value of the option. When jump risk exists, no arbitrage option value is no longer unique. Thus a delta hedging strategy and its associated hedging cost depends on how the risk is adjusted from the real world price process (4).
From the utility-based equilibrium theory, assuming y _< I is the risk aversion parameter, a risk-adjusted price process corresponding to (4) is (Lewis, 2002)
d[S. sub. t]/[S. sub. t] = (r - q - [[kappa].sup. Q][[lambda].sup. Q]) x dt + [sigma] xd[W. sup. Q.sub. t] + ([J. sup. Q] - 1) x d[[pi].sup. Q.sub. t],
where [W. sup. Q.sub. t] is a standard Brownian motion and [[pi].sup. Q.sub. t] is a Poisson counting process and log [J. sup. Q] is normally distributed with
[[mu].sup. Q.sub. J] = [[mu].sub. J] - (1 - [gamma])[[sigma].sup.2.sub. J]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where [K. sup. Q]. E([J. sup. Q] - 1). As an investor becomes more risk averse, the jump frequency and the expected magnitude of jumpsize are adjusted to larger quantities. In other words, the options are priced according to a larger (combined diffusion and jump) risk.
Our objective is to compute a hedging strategy under the assumption that a real world price dynamics (4) is given. To determine a delta hedge one needs to first choose a risk aversion parameter and then evaluate the risk adjusted lookback option values V(S, t). Choosing a risk aversion parameter is ad hoc in nature and much effort has been spent by economists to find appropriate risk aversion parameters. Alternatively, one can calibrate with the current market standard option prices to determine a risk adjusted process from these prices.
Under a Merton's jump model (5), analytic formula exists for density functions (Labahn, 2003) and standard option pricing. Assume that currently t = 0 and q(x) is a probability density function for x = [log. sub. ST] under (4). From Lewis (2002), the characteristic function for x is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
[omega] = [mu] - q - [[sigma].sup.2]/2 - [lambda][kappa].
Hence the probability density function for a Merton's jump model is given by
g(x) = 1/2[pi] = [[integral].sup.[infinity].sub.-[infinity]][[phi].sub. q](z)[e. sup.-izx]dz.
It can be easily shown that (Labahn, 2003)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
For computational results in this article, analytic formula for transitional density functions under a Merton's jump diffusion model (or Black-Scholes model) are used to compute delta hedging strategies and risk minimization hedging strategies. For delta hedging, the analytic formula for the transitional density function is used to compute values of lookback options based on (2). For risk minimization hedging strategies, analytic formulae for the transitional density function and standard option values are used to compute optimal hedging positions on a finite grid; the hedging strategies corresponding to independent simulations are computed using spline interpolations from the hedging positions on the grid.
When the risk aversion parameter [gamma] = 1, options are priced risk neutrally; the investors price the asset under the risk free interest rate. In our example, we assume that the jump frequency [lambda] = 20 percent, i. e. approximately a jump once every 5 years on average, a mean of -30 percent and a volatility of 15 percent for the logarithm of the jumpsize. Table 2 compares, under the risk neutral assumption, the effectiveness of delta hedging with the risk minimization hedging using the underlying. From Table 2, it can be observed that
* Both hedging strategies are more effective as one rebalances more frequently. However, due to jump risk, the improvement of more frequent rebalancing is less dramatic compared to that under the Black-Scholes model. Specifically, compared to annual rebalancing, monthly or biweekly rebalancing, under a Merton's jump diffusion model, improves the hedging performance only slightly further.
* Compared to delta hedging, risk minimization hedging using the underlying is significantly more effective in reducing risk under a jump diffusion model. Not surprisingly, risk minimization hedging incurs a larger average cost.
* For annual rebalancing, delta hedging can perform worse than not hedging at all as measured by the standard deviation of hedging error.
* The monthly risk minimization hedging using the underlying significantly reduces the risk, relative to an unhedged position.
If delta hedging is determined from a risk adjusted model which is calibrated from the option market and has a risk aversion parameter [gamma] < 1, the option is not priced risk neutrally. Thus the delta hedging strategy depends on how the option market or the hedger adjusts the jump risk; risk minimization hedging, on the other hand, is typically determined under a probability measure for the market price dynamics.
HEDGING USING STANDARD OPTIONS
With the rapid growth of derivative markets, it is now a common practice in the financial industry to use liquid vanilla options to hedge an exotic option. The S&P 500 index options traded on the exchange, for example, are natural instruments for hedging variable annuities with guarantees linked directly to or highly correlated to the S&P 500 index.
In this section, we assume that there is no transaction cost and compare risk minimization hedging using the underlying with risk minimization hedging using standard options under the Black-Scholes model as well as Merton's jump model; we discuss volatility risks in the next section.
To illustrate, we assume that hedging instruments are standard options with one year maturity; liquid options with a shorter maturity can similarly be used. We consider hedging using six options, three calls with strikes [100 percent, 110 percent, 120 percent] x St and three puts with strikes [80 percent, 90 percent, 100 percent] x St, and hedging using two at-the-money options (one call and one put with strike equal the current underlying value).
Table 3 compares, under a Black-Scholes model, monthly and annually rebalancing using the underlying with annual rebalancing using six options and two options, respectively. Table 4 presents a similar comparison but under a Merton's jump diffusion model (4). Note that, the value of the constant interest rate has minimal effect in hedging results and the change from r = 0.05 to r = 0.03 is inconsequential.
Tables 3 and 4 illustrate that, compared to hedging using the underlying, option hedging leads to better risk reduction. More specifically,
* Under a Black-Scholes model, monthly rebalancing using the underlying and annual rebalancing using two options have similar initial costs and average total costs. However, better risk reduction (measured either in standard deviation, VaR, or CVaR of the total risk) is achieved using two options compared to monthly hedging using the underlying. In addition, compared to hedging using two options, hedging using six options incurs slightly larger initial and average total costs but achieves significantly better risk reduction.
* Under a Merton's jump model, hedging using six options incurs slightly more initial and average total costs than hedging using two options. However, hedging using six options achieves better risk reduction than hedging using two options: contrast a CVaR(95 percent) value of 4.61 using six options with a CVaR(95 percent) value of 15.2 using two options. In addition, hedging using two options is much better than monthly rebalancing using the underlying in risk reduction; the initial cost and average total cost of option hedging is larger than that of hedging using the underlying. This result suggests that, in the presence of jump risk, option hedging leads to better risk reduction even though it may incur slightly larger costs.
HEDGING UNDER VOLATILITY RISKS
Computational results in the section "Hedging Using Standard Options" clearly suggest that standard options have potential use in hedging risks embedded in variable annuities. Specifically, Tables 3 and 4 suggest that hedging using standard options produces significantly greater risk reduction than using the underlying, particularly under a jump diffusion model.
Does this imply that one should hedge using options? To make this decision, we need to evaluate hedging strategies under more realistic market price evolution assumptions. In particular, when standard options are used as hedging instruments, quantifying risk and hedging performance depends on accurately modeling the market option price dynamics, in addition to the underlying dynamics. Moreover, one needs to consider other factors, e. g. transaction costs, liquidity risk, and default risk. In this section, we focus on evaluating hedging effectiveness under more realistic market price models which account for implied volatility risks.
Given the current market convention of quoting implied volatilities instead of option prices, accurately modeling evolution of market implied volatilities is necessary. Following market practice, implied volatility here is defined by inverting the BlackScholes formula from an option price.
Implied volatilities have been observed to display a curvature across moneyness and a term structure across time to maturity. In addition to modeling this static implied volatility structure, the evolution of the implied volatilities over time needs to be accurately modeled when standard options are used as hedging instruments. There has been active research in recent years on the evolution of the market implied volatilities over time based on historic implied volatility data, e. g. Cont and Fonseca (2002), Fengler, Hardle, and Villa (2003), and Zhu and Avellaneda (1997). Zhu and Avellaneda (1997) construct a statistical model for the term structure of the implied volatilities for currency options. Cont and Fonseca (2002) observe that, for major indices including S&P 500, the implied volatility surface changes dynamically over time in a way that is not taken into account by current modeling approaches, giving rise to "Vega" risk in option portfolios. They believe that option markets may have become increasingly autonomous and option prices are driven, in addition to movements in the underlying, also by internal supply and demand in the options market. They study the implied volatility time series for major indices, including S&P 500 from March 2, 2002 and February 2, 2003, and observe that:
* Implied volatilities are not static; they fluctuate around their means. The daily standard deviation of the implied volatility can be as large as a third of its typical value for out-of-the money options.
* The variance of daily log-variations in the implied volatility surface can be satisfactorily explained in terms of two or three principal components.
* The first principal component reflects an overall shift in the level of all implied volatilities which accounts for around 80 percent of the daily variance (Cont and Fonseca, 2002). The second principal component explains the opposite movement of the out-of-the-money call and put implied volatilities while the third principal component reflects change in the convexity of the volatility surface. In addition, there is a strong negative correlation between the index return and the change of the implied volatility level; the correlations between the underlying return and the other components of implied volatility changes are either significantly smaller or negligible.
Hence, in order to accurately quantify the hedging effectiveness when the hedging instruments consist of standard options, implied volatility risk needs to be suitably modeled. Unfortunately, under both the Black-Scholes model and Merton's jump diffusion model, implied volatilities are constant over time. This clearly is a substantial departure from the observed dynamics of the market implied volatilities.
How should the dynamics of the implied volatilities be modeled? To hedge a derivative contract, the typical approach is to assume a model for the underlying price evolution and derive implied volatilities from the assumed underlying price model; an underlying model can be estimated through calibration to the option market. Since both the Black-Scholes model and Merton's jump model lead to a static volatility surface, they are inadequate in modeling the observed stochastic implied volatilities. Among the typical models for underlying prices, only a stochastic instantaneous volatility underlying model can generate stochastic implied volatilities.
Unfortunately, the fact that instantaneous volatility is not directly observable presents many challenges for hedging and model estimation under an instantaneous volatility model.
* Firstly, it is difficult to adjust hedging positions according to an unobservable value.
* Secondly, it is not clear how to estimate an instantaneous volatility price model, from historical data, given that the instantaneous volatility is not observable; a stochastic instantaneous volatility underlying model can only be calibrated from liquid option prices. Moreover, the model obtained by calibrating to the market option prices gives the price model under a risk adjusted measure. As previously discussed, for hedging and risk management assessment we need a model for the real world market price dynamics.
* Thirdly, it is difficult to calibrate the current market option prices sufficiently accurately using a parsimonious stochastic volatility model with a small number of model parameters, e. g. Hestons model (Heston, 1993). In addition, there is no evidence that such a stochastic volatility model for the underlying is capable of modeling the evolution of the implied volatilities over time.
* Fourthly, even if a model calibrates the market prices by allowing a sufficient number of model parameters, e. g. a jump diffusion model with a local volatility function (Andersen and Andreasen, 2002), it is difficult to accurately model the implied volatility evolution for a long time horizon based on market calibration. A model that sufficiently calibrates the market implied volatilities today may give a poor fit for the implied volatilities tomorrow. This can be problematic for hedging and risk management purposes.
In addition to the cautionary remarks provided above, there are also arguments about whether an option pricing model derived from any underlying price model can adequately model option price evolution because of the autonomy of option market, e. g. Cont and Fonseca (2002). This suggests that it is reasonable and possibly better to consider a joint model for the underlying price and implied volatility evolutions.
Schonbucher (1999) jointly models, under a risk adjusted measure, the underlying and implied volatility evolution in which the instantaneous volatility and implied volatility both become state variables. No arbitrage restriction is established for the joint processes under a risk adjusted measure.
Since our focus is risk management and hedging for options embedded in variable annuities, we are interested in modeling the real world price evolution for both the underlying and hedging options. To evaluate hedging effectiveness of discrete dynamic hedging strategies for variable annuities, a joint discrete time model for real-world underlying and hedging instrument price changes between each rebalancing time is needed. In particular, this model needs to accurately describe the tails of the underlying price distribution and (mean reverting) stochastic implied volatility dynamics.
More specifically, we are interested in modeling the most liquid implied volatilities. We assume that the time to maturity of the option hedging instruments is fixed at one year (a shorter maturity can similarly be used) and far out-of-the money options are not used as hedging instruments due to liquidity considerations. Since change in the implied volatility level explains 80 percent of the daily variation in implied volatilities, as a first improvement, we model the at-the-money implied volatility evolution and assume that the ratios of implied volatilities to the at-the-money implied volatility are constant over time.
Let [[??].sub. t] denote the at-the-money implied volatility with a time to maturity of one year. Using an Ornstein-Uhlenbeck process, this at-the-money implied volatility fit is modeled as follows
d log([[??].sub. t]) = [??] x ([bar. log [[??].sub. t]] - log([[??].sub. t])) x dt + [[??].sub. vol] * d[Z. sub. t], (6)
where [Z. sub. t] is a standard Brownian, [bar. log [[??].sub. t]] denotes the long term average value (of the logarithm of the at-the-money implied volatility level). Figure 1 displays a sample path and distribution of the implied volatility [[??].sub. t] at time t = I year under the assumed model parameters; this model is used for at-the-money implied volatility in our subsequent computational investigation.
[FIGURE 1 OMITTED]
It is important to note that, by modeling implied volatilities directly, the standard option prices are directly given by the implied volatilities via the Black-Scholes formula; these liquid option values are no longer determined by the underlying price model. There are many advantages gained by computing a risk minimization hedging using standard options based on a model with implied volatilities directly as state variables. Firstly, the implied volatilities are directly observable from the market. This makes it possible to estimate a model from historic data and adjust hedging positions based on the implied volatilities. Secondly, by modeling the market implied volatility directly, calibration to market is automatically accomplished by setting the initial implied volatilities to the market implied volatilities. Thirdly, the hedging instruments are valued exactly using the Black-Scholes formula according to the market practice. For simplicity, we have modeled here only change in the level of the implied volatilities; but the approach can be extended (with additional computational complexity) to model the opposite movement of the implied volatilities for out-of-the money calls and puts and change in the convexity of the implied volatility curve.
Since the instantaneous volatility is not directly observable in practice, it is difficult to estimate a model with an instantaneous volatility as a state variable and adjust hedging positions based on this unobservable variable. Thus, when computing hedging strategies, we only assume a crude (constant) approximation [[theta].sub.0] of the instantaneous volatility (for example, a0 can be an estimation of the average of the instantaneous volatility over time). In other words, we compute risk minimization option hedging strategies under the joint model (7) below for the underlying price and the implied volatility [[??].sub. t]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (7)
where [W. sub. t] and [Z. sub. t] can be correlated in general. Note that the approach of determining option prices by modeling the underlying price evolution as described in the sections "Hedging Using the Underlying Asset" and "Hedging Using Standard Options" can be considered as a special case of a model (7) with [??] = [[??].sub. vol] = 0. In addition, jumps can be introduced in the mean reverting implied volatility dynamics as well.
Presently, for computational simplicity, we compute hedging strategies based on a joint model (7) assuming that Wt and [Z. sub. t] are independent. However, we evaluate hedging effectiveness under a joint model zoith both instantaneous volatility risk and implied volatility risk, and possibly correlation between these risks. Specifically, we evaluate hedging performance under the joint price model (8) below:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (8)
where [bar. log [theta]] a denotes the long term average value (of the logarithm of the instantaneous volatility), and [W. sub. t], [X. sub. t], and [Z. sub. t] can be correlated. Evaluating hedging effectiveness under this joint dynamics provides an assessment of the impact of ignoring correlation and instantaneous volatility risk in the hedging strategy computation.
We now compare effectiveness of the risk minimization hedging using the underlying with risk minimization hedging using six standard options. In addition to monthly hedging using the underlying, we compare the option hedging strategy computed by explicitly modeling the implied volatility risk in the risk minimization hedging computation (under column Option (6)-[??] with the risk minimization option hedging strategy for which the hedging positions at time [t. sub. k] are computed assuming that the implied volatility remains at the time [t. sub. k] level from [t. sub. k] to the maturity T (under column Option (6)-Reset); this is similar to computing a hedging strategy by recalibrating implied volatilities. We note that the risk minimization hedging Option (6)-[??] computes holdings on a three-dimensional grid, along the directions of the underlying, running max, and at-the-money implied volatility.
Tables 5 and 6 compare the hedging performance of these three strategies. In these computations, the initial implied volatilities are set to the Black-Scholes implied volatilities corresponding to the option prices computed under the underlying dynamics in the model (7); parameters for the model (7) are a = [??] = 0.6, [bar. log [theta]] = [bar. log [??]] = log (20 percent), with other parameters given explicitly in the table. The only difference in computational setup between Tables 5 and 6 is that the constant volatility [[theta].sub.0] in the model (7) is set to different values; thus the initial implied volatilities [[??].sub.0] are different as well. From Tables 5 and 6, we observe the following:
* Hedging using the underlying is sensitive to instantaneous volatility risk which is difficult to model; this is indicated by the different costs and risks under the column Underlying-[sigma] in Tables 5 and 6. In Table 5, hedging using the underlying leads to an under hedge of the payoff since a constant volatility of 15 percent, which is lower than the average of 20 percent, is used in the hedging computation. In Table 6, hedging using the underlying leads to an over hedge since a constant volatility of 22 percent, higher than the average of 20 percent, is used in the hedging computation.
* For hedging strategy Option (6)-Reset computed by resetting the implied volatility level at time [t. sub. k] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from [t. sub. k] to the maturity T, the hedging strategy and its performance is similarly sensitive to the initial implied volatility level. For example, hedging strategy Option (6)-Reset under-estimates the initial hedging cost in Table 5, while the initial hedging cost is over estimated in Table 6.
* When the implied volatility 4t is explicitly modeled in the risk minimization hedging framework, column Option (6)-[??], the hedging strategy computation takes the future implied volatility dynamics into consideration and the initial hedging cost is significantly less sensitive to the initial implied volatility level; note the striking similarities of the performance assessment under the column Option (6)-[??] for Tables 5 and 6. Moreover, we note that, compared to hedging using the underlying, hedging using options remains significantly more effective in risk reduction under both jump and volatility risks.
Table 7 displays the hedging performance when there is no instantaneous volatility risk (i. e. [[sigma].sub. vol] = 0 in (8) and instantaneous volatility is a constant). We emphasize that the implied volatility evolution is modeled in the option hedging strategy Option (6)-[??]. Comparing Table 7 with Table 5, the hedging results indicate sensitivities to the instantaneous volatility risk. It can be observed that hedging effectiveness of the risk minimization hedging using options under the implied volatility model (corresponding to the hedging strategy Option (6)-[??]) is relatively insensitive to instantaneous volatility risk. This suggests that hedging using standard options may potentially be an effective way of reducing instantaneous volatility risk, when implied volatility risk is properly modeled. Hedging using the underlying, on the other hand, is sensitive to the instantaneous volatility risk.
Tables 5 and 6 indicate that hedging using options by simply resetting the implied volatility level at the rebalancing time, i. e. Option (6)-Reset, either over - or underestimates the initial hedging costs. Figure 2 display the relative frequency (frequency divided by the number of simulations) of the increment cost ([U. sub. k+1] [[xi].sub. k+1] + [[eta].sub. k+1]) - ([U. sub. k]([t. sub. k+1])[[xi].sub. k] + [[eta].sub. k]). The top plot displays the relative frequency of the incremental cost for the hedging strategy Option (6)-[??] computed by explicitly modeling the implied volatility evolution; the bottom plot is for the hedging strategy Option (6)-reset which simply assumes that the future implied volatility stays at the current level at each rebalancing time. The hedging strategy, Option (6)-Reset, leads to a significantly larger average and variance of the incremental cost at each rebalancing time compared to the risk minimization hedging Option (6)-[??] for which the implied volatility risk is properly modeled.
[FIGURE 2 OMITTED]
Finally, Table 8 illustrates the hedging effectiveness of the three strategies evaluated under a correlation assumption between the change in log [[??].sub. t] and the Brownian innovation of the asset return, specifically corr([W. sub. t], [Z. sub. t]) = -0.6 in the model (8). Comparing Table 8 with Table 5, it can be observed that hedging effectiveness using standard options is not significantly affected by the correlation risk in terms of the standard deviation of the hedging error. However, VaR and CVaR become slightly smaller for this example.
Accurate quantification and robust hedging of the market risk embedded in a guaranteed minimum death benefit of a variable annuity contract is a new and challenging task for the insurance industry. This is mainly due to the exposure to the equity market risk, long maturity of the insurance contract, and the sensitivity of the benefit to the tails of the account value distribution. For this type of contracts, risk quantification and risk reduction require accurately modeling of the evolution of both the underlying account value and hedging instrument prices. In addition to a good model, effectiveness of a hedging strategy depends on the method from which hedging positions are determined. A delta hedging strategy is computed, independent of rebalancning frequency, based on option values under a risk adjusted measure. A risk minimization hedging strategy is computed to minimize risk based on a model for the real world price evolution and rebalancing specification. We analyze and compare these different hedging methods using either the underlying or standard options to hedge a lookback option embedded in variable annuity contracts for which the minimum death guarantee has a ratchet feature.
Due to sensitivity of such a variable annuity benefit to the tails of the account value distribution, jump risk and volatility risks need to be appropriately modeled. These additional risks, and the fact that dynamic hedging can only be implemented at discrete times with a limited choice of hedging instruments, suggest that hedging options embedded in variable annuities are hedging problems in an incomplete market. Thus it may be more appropriate to compute a hedging strategy to directly minimize a measure of the hedge risk given a set of trading times and a set of hedging instruments; this is the risk minimization hedging.
We evaluate and compare effectiveness of delta hedging with risk minimization hedging using the underlying under a Black-Scholes model as well as a Merton's jump diffusion model. We first assume that there is no volatility risk and illustrate that risk minimization hedging using the underlying is superior to delta hedging. In addition, monthly rebalancing risk minimization hedging using the underlying is relatively effective under a Black-Scholes model; the hedging effectiveness further improves as hedging portfolio is rebalanced more frequently. Moreover, under a Merton's jump model, risk reduction in both delta hedging and risk minimization hedging using the underlying is less effective. In particular, for both methods, improvement of biweekly hedging over monthly hedging is less significant.
Due to the increasing liquidity of standard options, we compare risk minimization hedging using standard options to that of using the underlying. We observe that the risk minimization hedging using standard options (even with only two at-the-money options) is significantly more effective than that of using the underlying, particularly when jump risk is considered.
Maturities of variable annuities are usually long; this makes modeling of relevant risks especially challenging. For hedging a long term derivative, volatility risk clearly cannot be ignored. Hedging using the underlying asset is susceptible to instantaneous volatility risk. When standard options are used as hedging instruments, hedging effectiveness needs to be evaluated under the implied volatility risk. Since the instantaneous volatility is not directly observable, it is difficult to model instantaneous volatility risk and implement hedging strategies which adjust hedging positions according to this unobservable variable. The typical approach of calibrating a model for the underlying evolution from the option prices is faced with the need for a complex model to match the current option prices and the need for a parsimonious model to reliably describe future option prices evolution. Moreover, the underlying price dynamics calibrated to the option market is under a risk adjusted measure rather than the real price dynamics necessary for risk quantification and hedging.
We compute a risk minimization hedging using standard options by explicitly modeling evolutions in the underlying as well as the at-the-money implied volatility. In our current investigation, the ratios of the implied volatilities of hedging instruments to the at-the-money implied volatility is assumed to be constant over time for simplicity. Since instantaneous volatility is not observable, we compute hedging strategy assuming a constant approximation to the instantaneous volatility in a model (7). By evaluating hedging performance under a joint model (8) of the underlying account value evolution, which includes instantaneous volatility risk, and implied volatilities, we illustrate that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options is effective in reducing lookback option risk embedded in variable annuity with a ratchet guaranteed minimum death benefit. In particular, unlike hedging using the underlying, which yields larger hedging error due to failure to model instantaneous volatility risk, effectiveness of the risk minimization hedging using standard options is relatively insensitive to the presence of instantaneous volatility risk. We also illustrate hedging performance when the change of the implied volatility is correlated to the change of the underlying.
Hedging variable annuities is a complex and challenging task. Investigation and analysis in this article are based on models which have been used to describe evolution of asset price with fat tails in the return distribution. For future investigation, it is important to evaluate how well a joint model (8) for the underlying and implied volatility can describe, in practice, the underlying market price and implied volatility evolution for a long time horizon. In addition to jump risk, volatility risks, other risks such as mortality risk, basis risk, and surrender risk also need to be properly analyzed. In a separate article Coleman, Li, and Patron (2006) we analyze sensitivity of risk minimization hedging to interest risk and compute the optimal risk minimization hedging strategy under both equity and interest risks.
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Heston, S. L. 1993, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies, 6: 327-343.
Hill, T. E. 2003, Variable Annuity with Guaranteed Benefits, Risk-Based Capital c-3 Phase ii, Technical Report Novmeber, Milliman USA Research Report.
Labahn, G. 2003, Closed form PDF Merton's Jump Diffusion Model, Technical Report, School of Computer Science, Unviersity of Waterloo, Waterloo, Ont. Canada N2L 3G1.
Lewis, A. 2002, Fear of Jumps, Wilmott Magazine, December: 60-67.
Longley-Cook, A. G. and J. Kehrberg, 2003, Efficient Stochastic Modeling Utilizing Representative Scenarios: Application to Equity Risks, Technical Report May, Milliman USA Research Report.
Mercurio, F. and T. C. F. Vorst, 1996, Option Pricing with Hedging at Fixed Trading Dates, Applied Mathematical Science, 3: 135-158.
Milesky, M. A. and S. E. Posner, 2001, The Titantic Option: Valuation of the Guaranteed Minimum Death Denefit and Mutual Funds, Journal of Risk and Insurance, 68: 93-128.
Patron, M. 2003, Risk Measures and Optimal Strategies for Discrete Hedging, Cornell University Ph. D. thesis.
Pelsser, Antoon, 2003, Pricing and Hedging Guaranteed Annuity Options via Static Option Replication, Insurance: Mathematics and Economics, 33: 283-296.
Pennacchi, G. G. 1999, The Value of Guarantees on Pension Fund Returns, Journal of Risk and Insurance, 66: 219-237.
Persson, S. A. and K. K. Aase, 1997, Valuation of Minimum Guaranteed Return Em - bedded in Life Insurance Products, Journal of Risk and Insurance, 64: 599-617.
Schal, M. 1994, On Quadratic Cost Criteria for Option Hedging, Mathematics of Operation Research, 19(1): 121-131.
Scheinerman, D. J. Kleiman, and D. Andrews, 2001, Variable Annuities with Minimum Death Benefit Guarantees, Technical Report, PricewaterhouseCoopers Principal Consultants-The Exchange, Actuarial News and Developments, 3(5).
Schonbucher, P. J. 1999, A Market Model for Stochastic Implied Volatility, Philosophical Transactions of the Royal Society, 357:2071-2092.
Schweizer, M. 1992, Mean Variance Hedging for General Claims, Annals of Applied Probability, 2: 171-179.
Schweizer, M. 1995, Variance-Optimal Hedging in Discrete Time, Mathematics of Operation Research, 20: 1-32.
Schweizer, M. 2001, A Guided Tour Through Quadratic Hedging Approaches, in: E. Jouini, J. Cvitanic, and M. Musiela, eds. Option Pricing, Interest Rates and Risk Management (Cambridge: Cambridge University Press), 538-574.
Zhu, Y. and M. Avellaneda, 1997, An E-ARCH Model for the Term Structure of Implied Volatility of FX Options, Applied Mathematical Finance, 4: 81-100.
T. F. Coleman is with the Faculty of Mathematics, University of Waterloo, Waterloo ON, N2L 3G1, Canada; Y. Kim is at the CTC Computational Finance Group, Cornell Theory Center, Cornell University; Y. Li is at the School of Computer Science, University of Waterloo, Waterloo ON, N2L 3G1 Canada; and M. Patron is at the Risk Capital, New York, NY, 10019, USA. The authors can be contacted via e-mail: tfcoleman@uwaterloo. ca, yhkim@tc. cornell. edu, yuying@cs. cornell. edu, and CPatron@riskcapital. com. The authors would like to thank their colleagues Peter Mansfield and Shirish Chinchalkar for many fruitful discussions. In addition, the authors would like to thank an anonymous referee whose detailed comments have improved the presentation of the article.
COPYRIGHT 2007 American Risk and Insurance Association, Inc. No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. Todos los derechos reservados.
Rafal Weron (Rafał Weron)
Burnecki, Krzysztof & Janczura, Joanna & Weron, Rafal, 2010. " Building Loss Models ," MPRA Paper 25492, University Library of Munich, Germany.
Krzysztof Burnecki & Joanna Janczura & Rafał Weron, 2010. " Building Loss Models ," SFB 649 Discussion Papers SFB649DP2010-048, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Krzysztof Burnecki & Joanna Janczura & Rafal Weron, 2010. " Building Loss Models ," HSC Research Reports HSC/10/03, Hugo Steinhaus Center, Wroclaw University of Technology.
Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," Papers 1010.1617, arXiv. org.
Agnieszka Janek & Tino Kluge & Rafał Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," SFB 649 Discussion Papers SFB649DP2010-047, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. " FX Smile in the Heston Model ," HSC Research Reports HSC/10/02, Hugo Steinhaus Center, Wroclaw University of Technology.
Borak, Szymon & Misiorek, Adam & Weron, Rafal, 2010. " Models for Heavy-tailed Asset Returns ," MPRA Paper 25494, University Library of Munich, Germany.
Szymon Borak & Adam Misiorek & Rafał Weron, 2010. " Models for Heavy-tailed Asset Returns ," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Szymon Borak & Adam Misiorek & Rafal Weron, 2010. " Models for Heavy-tailed Asset Returns ," HSC Research Reports HSC/10/01, Hugo Steinhaus Center, Wroclaw University of Technology.
Burnecki, Krzysztof & Weron, Rafal, 2010. " Simulation of Risk Processes ," MPRA Paper 25444, University Library of Munich, Germany.
Härdle, Wolfgang Karl & Burnecki, Krzysztof & Weron, Rafał, 2004. " Simulation of risk processes ," Papers 2004,01, Humboldt-Universität Berlin, Center for Applied Statistics and Economics (CASE).
Weron, Rafal, 2008. " Bezpieczeństwo elektroenergetyczne: Ryzyko > Zarządzanie ryzykiem > Bezpieczeństwo [Power security: Risk > Risk management > Security] ," MPRA Paper 18786, University Library of Munich, Germany, revised 2008.
Sznajd-Weron, Katarzyna & Weron, Rafal & Wloszczowska, Maja, 2008. " Outflow Dynamics in Modeling Oligopoly Markets: The Case of the Mobile Telecommunications Market in Poland ," MPRA Paper 10422, University Library of Munich, Germany.
Weron, Rafał & Burnecki, Krzysztof, 2004. " Modeling the risk process in the XploRe computing environment ," Papers 2004,08, Humboldt-Universität Berlin, Center for Applied Statistics and Economics (CASE).
Michael Bierbrauer & Stefan Trueck & Rafal Weron, 2005. " Modeling electricity prices with regime switching models ," Econometrics 0502005, EconWPA.
Szymon Borak & Wolfgang Härdle & Rafal Weron, 2005. " Stable Distributions ," SFB 649 Discussion Papers SFB649DP2005-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Ewa Broszkiewicz-Suwaj & Andrzej Makagon & Rafal Weron & Agnieszka Wylomanska, 2005. " On detecting and modeling periodic correlation in financial data ," Econometrics 0502006, EconWPA.
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You haven’t heard much from me lately, and I’ve been busy with a disproportionate amount of real-life stuff, but there are things to come yet.
I started working for a new company a few months ago. While this leaves me with less free time I can use for electronics and noodling, I’m fortunate enough to now be working professionally with things related to electronics, circuits, audio signal processing and microphones! I’m hoping this will have a positive effect on my DIY tinkeri ng as well smile emoticon
I’m actively working on the next update for my pet project, the StripboardCAD app. This time around I aim to improve workflow related to the parts list, entering of data/values etc. fueled by myself starting to use the tool actively and noticing a few short comings.
A few of my good musician friends have been trying out the “Workhorse” project I put together, and they’re so enthused about it that I’ve now built a few units for sale. That was a pain in the *behind*, and took much more time than I would have wanted, but hopefully it’ll come out well in the end. I even hear some plans about a demo video; a first for anything I’ve been involved with, and only (potentially) happening because I’m not doing it wink emoticon More to come on this soon (I hope).
I’m also revisiting the tap-tempo LFO stuff. While it has resulted in some very cool effects (I’m most happy with the tremolo), it has also disappointed on a few occasions (the phaser, to name one) due to overly noisy design etc. There’s two things to take away from this: 1) I need to learn more about proper grounding in digital effects, and how to lay stuff out on a vero/circuit board. 2) There are still things I would like the chip to do, like integrating a manual tempo adjust pot, a random wave form, and more options in the multiplier department (triplets etc).
And, of course, there are a number of other ideas I’m also playing with, that might result in something further down the road. Lo veremos.
I’ll try my best to post more news before the year is over, but just in case:
Merry christmas and a happy new year! Thanks for your support in 2017!
Updated: 5 December, 2017 — 13:36
I’ve just about finished the first major update to StripboardCAD. What remains now is a bit of testing, and the App Store submission process.
This update is all about fixing an annoying issue with some layouts missing some of the parts list; an issue I discovered right after the initial release. Along with the bug fix I’ve rewritten most of the layout generation engine, adding some more options and general tweaking.
As a bonus I’ve added a few requested features that were easy to implement, amongst them the ability to create layouts with the component values printed directly on (rather than the label).
I’ve added a change log page for keeping track of what’s being added as time goes by. Also, do let me know if you have issues, requests of ideas of improvement
Updated: 16 August, 2017 — 02:43
This one has been cooking for a while. An overdrive in the Centaur genre with a few changes and modifications.
Full page with schematic, layout and lots of more or less interesting explanations: Workhorse
Updated: 8 August, 2017 — 00:09
As quite a few of you know I spent most of my time last year working on a vero layout app for iPad. I was initially hoping to finish it up by Christmas, but obviously I missed that mark. That’s the bad news.
The good news is I’m still working diligently getting it done (I guess that might actually be bad news for those of you hoping I’d have more layouts ready instead). I’ve got the famous 90% ready, and only missing the last 10% that always end up taking the most time.
Here’s a few screenshots so you can get an idea where I’m at right now.
As you can see I did a quick mock-up of the Korg SDD-3000 layout, and it looks quite nice. But you’ll also notice there are a few things missing still.
All the basic graphics work is done, with only minor tweaks remaining. It’s vector based, scrolls and zooms nicely, and looks great when exported to PDF. There’s the component picker on the left and the parts list on the right, and you basically just tap, draw and move stuff around as you wish.
There’s a basic menu for entering meta data, like layout title, author etc. and this info will of course end up on the exported layout along with the layout image and parts list.
As my goal for version 1 is to be able to replicate my own layouts the way I have them drawn now, I’ve left out things like pots, switches and wire. That’s not to say they won’t be included, but not in version 1 (or I’ll never finish this thing). What I’m going to have instead is an “off-board” menu where textual representation of these things can be entered (like my layouts today).
The export menu is also under construction still. I’m hoping to be able to generate both PDFs and PNGs, but may cut it short with just PDF to get started.
Also, not visible here, is a file manager back-end that needs to be in place before release. I have something very basic right now that needs to be replaced with something proper. iCloud-support may or may not make it for this version, but I’m sure to add it eventually since it’s a feature I want to use myself.
Plans now are to complete the missing parts and try to get this thing up on the App Store somehow. I think there’s a beta-program as well, but again, that’s another few developer documents/books I need to consume to have it all figured out. We’ll see, but in the event I’ll have a beta run I’ll be happy with all the help I can get.
And so I have a somewhat awkward question for all of you. Assuming I’ll get this thing out the door, I need to put a price tag on it. I’m not totally comfortable charging money for the things I provide here (I like giving stuff away), but then again I can certainly also use what funding I would get and put it back into future development and other related projects. I have probably spent several hundred hours on this so far (most of it being a lot of fun) and would of course like to see other people getting some use from this, but in the end I did build it for myself first and foremost. What do you people think? What would you be realistically willing to spend on an app like this? From asking around amongst friend I get everything from $1 to $100.
De todas formas. Hope you all have a great 2017!
Updated: 13 January, 2017 — 01:17
A repository for q/kdb+ programs. For more details on q/kdb+ visit www. kx. com .
Keywords: Binomial Model, Black-Scholes Equation, Implied Volatility, Smile, Options Pricing, Delta Hedging, Risk Neutral Distribution, Physical Distribution, Statistics, Portfolio Tracking, Tickerplant, Web Socket, JQuery UI.
Portfolio Tracker version 0.4 is now available under /portfolioTracker directory.
New features in version 0.4:
Enhancements to support strategy backtesting by replaying historical tick data.
Historical tick data is sourced from http://www. netfonds. no. Of course you can use other sources, but may require editing of timersvc. q under /histTickData directory.
Delta Hedging Simulator is now available under /deltaHedging directory.
Binary and Barrier Options Pricing.
Here we tried to price two types of options. The Higher and the No Touch which are available on binary. com website.
Volatility are considered to be constant. Here I used Implied Volatility calibrated based on price available from binary. com website.
1-month LIBOR is used as risk free rate.
FX rates data is sourced from Yahoo Finance.
Communication between back end and front end is done using Web Socket.
Derman, Emanuel, and lraj Kani. "The Ins and Outs of Barrier Option: Part 2." Derivatives Quarterly, Winrcr 96 URL: http://www. emanuelderman. com/media/insoutbarriers2.pdf
Implied Risk Neutral Distribution.
Computation of Implied Risk Neutral Distribution by recovering from index option prices.
Here I use Nikkei 225 Options data available from Osaka Stock Exchange (OSE) website.
Data source: http://www. jpx. co. jp/markets/statistics-derivatives/daily/index. html
Shimko, David C(1993): "Bounds of Probability", Risk, 6, 33-7
Demonstrate the use of feed handler, tickerplant as well as tick subscriber.
Market data is sourced from Yahoo Finance. Please note that it is a delayed tick data.
A feed handler based on Random Walk Simulation is also available so that application can be tested during market close. This simulator can be found under feedHandler/randomWalkSim directory. The file is feedsimR. q.
PnL is calculated based on the difference between average bought/sold price and last price.
Charts are rendered using Google Chart Api.
Web Socket is used to push data to GUI.
Physical Distribution Estimation.
An estimation of physical distribution based on historical data using Gaussian Kernel.
Here I am using Nikkei 225 daily historical data for year 2017. This data was downloaded from Yahoo Finance.
BS Model Viewer.
A graphical viewer for Black-Scholes model for European Call Option.
The GUI allows user to change parameters and see how the Greek letters change.
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