## Modular Pricing of OptionsThe 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. |

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### Contents

Introduction | 1 |

12 Constructing Characteristic Functions | 5 |

13 Economic Interpretation of CFs | 10 |

14 Examination of Existing Option Models | 15 |

15 Equivalence of CFs to PDEs | 19 |

Modular Pricing of Options MPO | 25 |

22 Stochastic Volatilities | 28 |

222 Square Root Process | 30 |

Extensions of MPO to Exotic Options | 99 |

32 Barrier Options | 100 |

322 Two Special Cases | 103 |

323 Numerical Examples | 109 |

33 Lookback Options | 115 |

332 Pricing Formulae with Stochastic Factors | 116 |

34 Asian Options | 122 |

342 The BlackScholes World | 123 |

223 OrnsteinUhlenbeck Process | 36 |

224 Double Square Root Process | 49 |

23 Stochastic Interest Rates | 56 |

232 Square Root Process | 58 |

233 OrnsteinUhlenbeck Process | 62 |

234 Double Square Root Process | 65 |

24 Random Jumps | 68 |

242 Pure Jumps | 72 |

243 Lognormal Jumps | 74 |

244 Pareto Jumps | 77 |

25 Integrating the Modules | 80 |

252 Pricing Kernels for Options and Bonds | 87 |

SI versus SV | 88 |

26 Appendices | 93 |

Derivation of the CFs with Double Square Root Process | 96 |

343 Asian Options in a Stochastic World | 128 |

344 Approximations for Arithmetic Average Asian Options | 132 |

345 A Model for Asian Interest Rate Options | 135 |

35 Correlation Options | 138 |

352 Exchange Options | 142 |

353 Quotient Options | 146 |

354 Product Options | 147 |

36 Other Exotic Options | 149 |

37 Appendices | 151 |

Derivation of the CFs for Correlation Options | 154 |

Conclusions | 157 |

List for Notations and Symbols | 161 |

163 | |

### Common terms and phrases

90 Call Asian options asset price average Asian options barrier options benchmark Black-Scholes formula Brownian motion calculated call option characteristic functions closed-form solution correlated with stock correlation options denote derived double square root dr(t dx(t exchange options exotic options formula for options Fourier analysis Fourier inversion framework geometric average given Heston model implied volatilities knock-out options ln(l lookback options martingale mean-reverting O-U process modular pricing moneyness obtain option pricing formula option pricing models Ornstein-Uhlenbeck process OTM options Panel parameters Pareto jumps payoff Poisson process Pr(m probability product options pure jumps put options quotient options random jumps risk-neutral risk-neutral measure Schobel and Zhu Section specified square root process squared volatility stochastic factors stochastic interest rates stochastic volatility models stock price process stock returns strike price subsection underlying assets valuation vanilla options variance Vasicek Vasicek model zero

### References to this book

Integrated Market and Credit Portfolio Models: Risk Measurement and ... Peter Grundke Limited preview - 2008 |