Since the appearance of seminal works by R. Merton, and F. Blackand M. Scholes, stochastic processes have assumed an increasinglyimportant role in the development of the mathematical theory offinance. This work examines, in some detail, that part of stochasticfinance pertaining to option pricing theory. Thus the exposition isconfined to areas of stochastic finance that are relevant to thetheory, omitting such topics as futures and term-structure.This self-contained work begins with five introductory chapterson stochastic analysis, making it accessible to readers with little orno prior knowledge of stochastic processes or stochastic analysis.These chapters cover the essentials of Ito's theory of stochasticintegration, integration with respect to semimartingales, Girsanov'sTheorem, and a brief introduction to stochastic differentialequations.Subsequent chapters treat more specialized topics, includingoption pricing in discrete time, continuous time trading, arbitrage,complete markets, European options (Black and Scholes Theory),American options, Russian options, discrete approximations, and assetpricing with stochastic volatility. In several chapters, new resultsare presented. A unique feature of the book is its emphasis onarbitrage, in particular, the relationship between arbitrage andequivalent martingale measures (EMM), and the derivation of necessaryand sufficient conditions for no arbitrage (NA).{\it Introduction to Option Pricing Theory} is intended forstudents and researchers in statistics, applied mathematics, business,or economics, who have a background in measure theory and havecompleted probability theory at the intermediate level. The worklends itself to self-study, as well as to a one-semester course at thegraduate level.