Probability Theory in Finance - A Mathematical Guide to the Black-Scholes Formula
The primary aim of this book is to offer a rigorous probabilistic treatment to more delicate topics of financial mathematics, exemplified by the Black-Scholes formula: a well known and extremely efficient tool to analyze a wide range of options prizes. The author builds up the necessary finance theory and mathematics simultaneously. Having started with concepts such as money, interest rates, markets, hedging and arbitrage, he moves on to the basics of measure theory (σ-fields, filtrations, measures, measurability and convergences, expectations, independence, product measures and conditioning). Short excursions both into finance and mathematical analysis are offered and inserted whenever it might be convenient for the reader and natural in the course of the presentation. The call options and hedging on the financial side and the continuous and convex functions and Riemann integration on the side of analysis may serve as examples. The central limit theorem, discrete martingales, the Wiener process and its exponential, the elements of stochastic integration (including the Girsanov change of measure) are treated rigorously to prepare the reader for the final goal of the book, which is the Black-Scholes formula. The considerable collection of exercises with solutions available has an important role in the neatly and lively written text that may be warmly recommended to undergraduates and graduate students in mathematics and finance.