Introduction to the Mathematics of Finance
This very nice monograph is intended as an introductory course to mathematical finance. It assumes a basic knowledge of probability theory, discrete and continuous time martingales, Brownian motion and stochastic calculus. The textbook consists of five chapters and four appendices. Chapter 1 is an introduction to the basic terms from the financial market, with examples and some assumptions to simplify the presentation. In chapter 2, the Cox-Ross-Rubinstein binomial model is described and the arbitrage free prices for both European and American contingent claims are derived. The idea of European option pricing is extended to a general finite market model in chapter 3, where the first and second fundamental theorems of asset pricing are formulated and proved. In chapter 4, the Black-Scholes model is introduced, for which pricing and hedging of European and American contingent claims are developed, including the famous Black-Scholes option pricing formula. In chapter 5, the multi-dimensional Black-Scholes model is defined, which is considered as a continuous analogue of the discrete finite market model, and the fundamental theorems of asset pricing are extended to a class of continuous models. In the appendices, basic results from conditioning, discrete and continuous time martingales and stochastic calculus for Brownian motion are reviewed. The text is clearly written and well-arranged and most of the results are proved in detail. Each chapter is completed with exercises, which makes the textbook very comprehensive.