## Applications of Fourier Transform to Smile Modeling: Theory and ImplementationThis 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. |

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### Contents

1 | |

Characteristic Functions in Option Pricing | 21 |

Stochastic Volatility Models | 45 |

Numerical Issues of Stochastic Volatility Models | 77 |

Simulating Stochastic Volatility Models | 113 |

Stochastic Interest Models | 135 |

Poisson Jumps | 153 |

Lévy Jumps | 173 |

Integrating Various Stochastic Factors | 203 |

Exotic Options with Stochastic Volatilities | 223 |

Libor Market Model with Stochastic Volatilities | 273 |

319 | |

327 | |

### Other editions - View all

Applications of Fourier Transform to Smile Modeling: Theory and Implementation Jianwei Zhu No preview available - 2009 |

Applications of Fourier Transform to Smile Modeling: Theory and Implementation Jianwei Zhu No preview available - 2012 |

### Common terms and phrases

algorithm applied approach Asian options barrier options Black-Scholes model Brownian motion calculate calibration call option caplet Carr Chapter characteristic function closed-form solution complex number compute correlation corresponding deﬁned denote deterministic distribution double square root dV(t dx(t dynamics equation exotic options expected value ﬁrst forward measure Fourier transform gamma given hedging Heston model implied volatilities integration jump model L´evy jump L´evy process Lévy Lévy process Libor LogN Madan martingale mean-reverting Ornstein-Uhlenbeck process mean-reverting square root measure Q moment-generating function numeraire option pricing option pricing formula option pricing models Ornstein-Uhlenbeck process parameters Poisson jumps Poisson process random variable risk risk-neutral measure risk-neutral process scheme simulation skew smile modeling spot volatility square root process stochastic factors stochastic interest rates stochastic volatility models stock returns strikes swap rate swaptions theorem time-change V(th valuation variance variance-gamma process volatility smile zero-coupon bond