Financial mathematics is going through a period of intensive development, particularly in the area of stochastic analysis. This timely work presents a comprehensive, self-contained introduction to stochastic financial mathematics. It is based on lectures given at Moscow State University, ""Stochastic Analysis in Finance"", and comprises the basic methods and key results of the theory of derivative securities pricing in discrete financial markets.The following elements: martingales, semimartingales, stochastic exponents, Ito's formula, Girsanov's theorem, and more, are used to characterize notions such as arbitrage and completeness of financial markets, fair price and hedging strategies for options, forward and futures pricing, and utility maximization. Limiting transition from a discrete to continuous model with derivation of the famous Black-Scholes formula is shown. The book contains a wide spectrum of material and can serve as a bridge to continuous models. It is suitable as a text for graduate and advanced graduate students studying economics and/or financial mathematics.