Option Theory takes the reader from first principles to the frontiers of modern finance theory. The book is aimed at busy financial engineers at all levels, providing formulas and techniques that can be readily applied to real life problems; yet the theoretical basis of the subject is explored in detail so that the book will also appeal to students and researchers. Written in a clear and accessible manner, the author covers the various approaches to option pricing: risk neutral expectations by integration, trees, analytical and numerical solutions of partial differential equations and Monte Carlo methods, demonstrating the close relationship between them. Structured into four parts, the mathematical tools used in the first three parts of the book are intermediate level "engineer's mathematics": differential and integral calculus, elementary statistical theory and simple partial differential equations. In Part Three, the techniques are systematically applied to all the standard exotic options encountered in the equity, foreign exchange and commodity markets. It is shown that the exotics are not a large random collection of unrelated instruments, but a few families which can be simply analysed using the techniques developed in Parts One and Two. Part Four provides a course in stochastic calculus that is specifically tailored to finance theory and designed for readers with some previous knowledge of options. It provides an active working knowledge of the subject and includes coverage of: Martingales. Stochastic differential equations. Stochastic integration. The Feyman Kac theorem. Stochastic control. Local time. Girsanov's theorem. The axiomatic approach to option theory using stochastic calculus is compared in detail to the simpler and more intuitive approach using classical statistics, which was used in the first three parts of the book. The analysis clearly shows where stochastic calculus provides valuable insights and advances, and where it is mere window dressing. This is a no-nonsense professional book which demystifies and simplifies the subject, and which will appeal to both practitioners and students.