Martingale Methods in Financial Modelling

The book provides a comprehensive, self-contained and up-to-date treatment of the main topics in the theory of option pricing. The first part of the text deals with simple discrete models of financial markets, including the Cox-Ross-Rubinstein binomial model. No knowledge of probability and stochastic processes is assumed at this stage, while most of the concepts from modern mathematical finance are explained at a very elementary mathematical level. The passage from the discrete to the continuous market models, done in the Black-Scholes model setting, assumes familiarity with basic ideas and results from stochastic calculus such as Wiener process and Ito formula; however, an appendix containing all the necessary results is included. The Black-Scholes setting is later generalized to cover standard and exotic options involving several assets and/or currencies. Numerous examples of exotic options are analysed. An outline of a general theory of arbitrage pricing is presented. A very substantial part of the text is devoted to term structure modelling and to the pricing of interest rate options. The HJM framework is discussed in detail. Models based on the forward LIBOR and forward swap rates are introduced. The main emphasis is on models that can be made consistent with the market pricing practice.