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.
Basic concepts and objects of a financial market The elements of discrete stochastic analysis A stochastic model for a financial market. Arbitrage and completeness Pricing European options in complete markets. The binomial model and the Cox-Ross-Rubinstein formula Pricing and hedging American options in complete markets Financial computations on a complete market with the use of nonself-financing strategies Incomplete markets. Pricing of options and problems of minimizing risk The structure of prices of other instruments of a financial market. Forwards, futures, bonds The problem of optimal investment The concept of continuous models. Limiting transitions from a discrete market to a continuous one. The Black-Scholes formula Appendix 1 Appendix 2 Appendix 3 Hints for solving the problems Bibliography Subject index.