## Mathematical Modeling and Methods of Option PricingFrom the unique perspective of partial differential equations (PDE), this self-contained book presents a systematic, advanced introduction to the Black-Scholes-Merton's option pricing theory.A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs. In particular, the qualitative and quantitative analysis of American option pricing is treated based on free boundary problems, and the implied volatility as an inverse problem is solved in the optimal control framework of parabolic equations. |

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### Contents

Risk Management and Financial Derivatives | 1 |

ArbitrageFree Principle | 9 |

Binomial Tree Methods Discrete Models of Option Pricing | 25 |

Brownian Motion and I to Formula | 55 |

European Option Pricing BlackScholes Formula | 73 |

nite Difference Method | 100 |

American Option Pricing and Optimal Exercise Strategy | 113 |

MultiAsset Option Pricing | 201 |

PathDependent Options | 247 |

Implied Volatility | 311 |

Bibliography | 323 |

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### Common terms and phrases

American call option American option price American put option arbitrage-free arbitrage-free principle average Asian option barrier options Black-Scholes equation Black-Scholes formula boundary conditions Brownian motion call option price call-put parity Cauchy problem continuation region defined denote derivatives discrete dividend rate domain early exercise European call option European option European option pricing European put option expiration date fixed strike price floating strike price free boundary problem geometric average Asian implied volatility investor Ito formula Lemma lookback options mathematical model measure Q multi-asset options optimal exercise boundary option as example option with fixed option with floating parabolic equation Parisian option path-dependent options payoff function portfolio premium Proof put option price reset options risk risk-free interest rate risk-neutral measure risky asset satisfies solution solve stochastic stock price stopping region Substituting terminal condition Theorem transformation underlying asset price value problem vanilla option variational inequality