Stochastic Finance: An Introduction in Discrete Time
This text on probabilistic methods in finance is intended for graduate students in mathematics with some background in probability: stochastic models in discrete time are considered.
The exposition has two parts. In the first part (Chapters 1-4), the one-period model of a financial market is considered. Basic principles of mathematical finance are explained: portfolio, market efficiency, arbitrage, martingale measure, options, contingent claims, perfect hedge, complete and incomplete market. The mathematical theory of utility is then presented, and used for portfolio optimisation and risk measuring. The second part (Chapters 5-10) deals with multi-period models of financial market that are characterised by stochastic processes in discrete time. Chapter 5 is of fundamental importance and the true core of the book: dynamic arbitrage theory, based on martingales and equivalent martingale measures; European contingent claim and the problem of pricing of this financial derivative; the Cox, Ross and Rubinstein binomial model; limit theory, leading to the well-known Black-Scholes pricing formula in continuous time. The remaining chapters are devoted to various hedging strategies. An appendix summarises some results from probability, convex and functional analysis.
This is an excellent textbook. Since considerations in discrete-time stochastic financial models are simpler, the exposition proceeds quickly to key problems in the theory of pricing and hedging of financial derivatives. However, the authors formulate their models on general probability spaces, so the text also captures some interplay between probability theory and functional analysis.