Applications of Fourier Transform to Smile Modeling

The sound modeling of the smile effect is an important issue in quantitative finance as, for more than a decade, the Fourier transform has established itself as the most efficient tool for deriving closed-form option pricing formulas in various model classes. This book describes the applications of the Fourier transform to the modeling of volatility smile, followed by a comprehensive treatment of option valuation in a unified framework, covering stochastic volatilities and interest rates, Poisson and Levy jumps, including various asset classes such as equity, FX and interest rates, as well as various numberical examples and prototype programming codes. Readers will benefit from this book not only by gaining an overview of the advanced theory and the vast range of literature on these topics, but also by receiving first-hand feedback on the practical applications and implementations of the theory. The book is aimed at financial engineers, risk managers, graduate students and researchers.

Zusammenfassung
This book addresses the applications of Fourier transform to smile modeling. Smile effect is used generically by ?nancial engineers and risk managers to refer to the inconsistences of quoted implied volatilities in ?nancial markets, or more mat- matically, to the leptokurtic distributions of ?nancial assets and indices. Therefore, a sound modeling of smile effect is the central challenge in quantitative ?nance. Since more than one decade, Fourier transform has triggered a technical revolution in option pricing theory. Almost all new developed option pricing models, es- cially in connection with stochastic volatility and random jump, have extensively applied Fourier transform and the corresponding inverse transform to express - tion pricing formulas. The large accommodation of the Fourier transform allows for a very convenient modeling with a general class of stochastic processes and d- tributions. This book is then intended to present a comprehensive treatment of the Fourier transform in the option valuation, covering the most stochastic factors such as stochastic volatilities and interest rates, Poisson and Levy ´ jumps, including some asset classes such as equity, FX and interest rates, and providing numerical ex- ples and prototype programming codes. I hope that readers will bene?t from this book not only by gaining an overview of the advanced theory and the vast large l- erature on these topics, but also by gaining a ?rst-hand feedback from the practice on the applications and implementations of the theory.

Inhalt
Option Valuation and the Volatility Smile.- Characteristic Functions in Option Pricing.- Stochastic Volatility Models.- Numerical Issues of Stochastic Volatility Models.- Simulating Stochastic Volatility Models.- Stochastic Interest Models.- Poisson Jumps.- Lévy Jumps.- Integrating Various Stochastic Factors.- Exotic Options with Stochastic Volatilities.- Libor Market Model with Stochastic Volatilities.