From 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.

Contents:

• Risk Management and Financial Derivatives
• Arbitrage-Free Principle
• Binomial Tree Methods — Discrete Models of Option Pricing
• Brownian Motion and Itô Formula
• European Option Pricing — Black–Scholes Formula
• American Option Pricing and Optimal Exercise Strategy
• Multi-Asset Option Pricing
• Path-Dependent Options (I) — Weakly Path-Dependent Options
• Path-Dependent Options (II) — Strongly Path-Dependent Options
• Implied Volatility

“It offers an excellent coverage of the PDE methods in financial mathematics, including free boundary problems (for American options) and optimal control, i.e. inverse problems (for implied volatility) ... I recommend this book most enthusiastically to every practitioner or student of financial mathematics. Also, the book is suitable as a textbook for a graduate course in financial mathematics, especially if complemented with some other text for a wider coverage.”