## PDE and Martingale Methods in Option PricingThis book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform. |

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A fantastic, in-depth, well written book. I am surprised it is not more popular! Gives equal weighting to the PDE and martingale approaches in finance; well spaced, glossy, well referenced, etc.

### Contents

1 | |

15 | |

3 Continuoustime stochastic processes | 97 |

4 Brownian integration | 138 |

5 Itˆo calculus | 167 |

uniqueness | 203 |

7 BlackScholes model | 218 |

existence | 257 |

11 American options | 389 |

12 Numerical methods | 402 |

13 Introduction to Levy processes | 429 |

14 Stochastic calculus for jump processes | 497 |

15 Fourier methods | 541 |

16 Elements of Malliavin calculus | 577 |

a primer in probability and parabolic PDEs | 598 |

691 | |

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### Common terms and phrases

American option analogous approximation arbitrage arbitrage price Asian options assume assumption binomial bounded variation Brownian motion Call option Cauchy problem characteristic exponent claim follows classical coefficients compound Poisson process compute condition consider continuous convergence Corollary d-dimensional defined Definition denote density derivative distribution EMM Q equivalent Example exists finite Fourier fundamental solution Heston model implied volatility Itô Itó formula jumps Lemma Lévy measure Lévy process linear Lipschitz continuous martingale matrix non-negative notation numeraire option pricing parabolic parameters payoff Poisson process positive constant probability space Proposition prove Put option quadratic variation random variables recall Remark replicating result risk-neutral price risky asset Section self-financing strategy sequence stochastic integral stochastic process stochastic volatility underlying asset uniqueness verify