The aim of the book, as it states in the preface, is “…to provide basic stochastic analysis techniques for mathematical finance.” Indeed, this is exactly what this book provides, a basic introduction of stochastic analysis to mathematical finance. I say basic because there is much more to stochastic analysis that is needed in advanced mathematical finance than what can be found in this book. Generally, for advanced mathematical finance (where one would even use stochastic analysis), one would have to know measure theory and stochastic calculus. This book focuses more on the applied side of some more important theorems from stochastic analysis. Due to this approach the book is very suitable for practitioners who just need to understand the main connections between some financial concepts and stochastic analysis. One would have to have some background in finance to be able to understand the examples, as the author does not explain what bonds and options are, nor does he explain some of the computations related to them. This can serve as a short “stochastic finance” textbook. Due to the lack of exercises I believe that this would be more suitable for graduate students with appropriate background in stochastic analysis.
The book is structured in four parts: introduction, the discrete case, the continuous case and applications. The main points of probability theory are very briefly outlined in chapter 2, which covers topics from a “flipping a coin” to conditional expectations and the central limit theorem. This chapter must be taken as a little reminder of the topics mentioned; it does not provide detailed results. The assumption, clearly, is that the reader already knows enough probability theory to read the rest of the book.
The introduction continues in chapter 3, which deals with discrete-time stochastic processes. Filtrations, along with the notion of filtered space, are presented very briefly. The author gives couple of nice intuitive examples to support the theoretical concepts presented. Next, random walks, discrete Markov chains and martingales are explained. An example is presented with each topic. These examples are excellent for readers who don’t have much experience in applying stochastic theory to financial problems. The chapter ends with two larger examples, one on option pricing with binomial models and the other on binomial interest rate models, both adjusted to a discrete case.
Chapter 4 introduces continuous-time stochastic processes. The first process covered is Brownian motion followed by the Poisson process. Main definitions and assumptions are explained intuitively. At this point almost all the necessary background is covered, so the stochastic calculus is introduced. This is extremely important for mathematical finance and it requires a very intuitive presentation as well as the rigorous discussion. As always, the non-differentiability of Brownian notion is the starting point, along with the stochastic integration. Next, stochastic differential equations and Itô’s lemma are covered. The author gives a continuous notion of the interest rate model presented as a discrete scenario in previous chapter. The famous Black-Scholes formula is introduced.
Chapter 6 focuses on more difficult topics that are necessary for mathematical finance. These include the Feyman-Kac formula and the Girsanov theorem. Again the author gives detailed examples applicable to finance theory. The rest of the book deals with specific examples from the insurance field.
Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.