Martingale Methods in Financial Modelling

Mathematical finance is a rather new area of research in mathematics, but nevertheless a popular one. This text is just but one example of the current level of sophistication in the application of mathematical techniques to the area of finance. Today not only students and academics would read a book of such caliber but also the practitioners in various financial institutions. I see this book to hold its place quite high with practitioners as bigger financial institutions really do practice the financial instruments explained in the book.

Martingale Method in Financial Modelling is the 2^nd edition of a well-received book. It brings new additions and updates as well as some completely new chapters, such as the one on Volatility Risk. The authors did an excellent job of providing updated references to current work in each of the areas presented in the book, so that the more curious reader can find even more excitement outside of this book.

One important claim of this (quite large) book is that, as the authors mention in the preface, it does not require any knowledge of finance. How true this is really depends on whether you are a mathematician who is interested in financial modeling (and just beginning to investigate problems in finance) or a mathematician with some finance background. Nevertheless, authors do explain financial instruments to a certain extent in the first chapter, and then later on in each chapter as needed. However, I would recommend that one should first read John Hull’s Options, Futures, and Other Derivatives or an equivalent in order to really get a grasp on all these different financial instruments and how they are traded or used. There are almost no examples in the text except some tedious ones in the first chapters, which is why it is even more imperative that the audience has some knowledge of finance in order to have the clear picture in the head of all the possibilities that can occur when modeling financial instrument. For a good experience in reading this book one should have a good knowledge of probability and stochastic calculus. Authors do provide an appendix that covers some stochastic calculus.

The book is divided into two parts: Part I deals with Spots and Futures Markets and Part II with Fixed-income Markets. It starts lightly, trying to make the reader feel comfortable in her chair. Chapter 1 and 2 give preliminaries on financial instruments and discrete time modeling based on arbitrage framework. Chapter 3 jumps on to continuous time and the famous Black-Scholes model. Also we go back to 1900s and the famous Bachelier option pricing formula. Chapter 4 provides valuation formulas for different kinds of foreign currency and equity options. Chapter 5 covers the “flexible” American options in detail and then continues onto exotic options in chapter 6. The issue of modeling volatility has been a big subject over the years. The authors decided, in this new edition, to add a chapter that deals with problems of modelling volatility risk. Chapter 7 deals with implied volatilities, local volatility models as well as stochastic volatility models accompanied by the presentation of the dynamics of volatility surfaces. Part I ends with continuous time security markets covering basic results in a continuous time setting.

Part II is devoted to fixed income markets. This is also a great area of mathematical finance. Naturally, the section starts the reader off with a discussion of interest rates and related concepts. Chapter 9 gives fast descriptions of such fixed income structures and instruments as zero-coupon bonds, forward interest rates, FRAs, interest rate futures, etc. while the chapter finishes off with the presentation of the stochastic models of bond prices. Chapter 10 gives short-term rate models such as single-factor and multi-factor models. Heath, Jarrow and Morton, i.e. HJM approach to term structure modelling is presented in chapter 11. Chapter 12 continues onto arbitrage-free market LIBOR models. The book finishes off with modelling derivatives securities presented in at least two different markets or economies, i.e. the cross-currency derivatives.

Martingale Methods in Financial Modelling is an authoritative text which gives a lot of insight into financial instruments and current modeling practices. The book based on lecture notes and one of its purposes is to serve as a classroom text. Certainly there is plenty of material for a course; however, there are no exercises at the end of the chapters, which makes it hard for students to test their knowledge or to practice the methods learned.

At first glance the book gives an impression of a dry and cold presentation of financial modeling. However, I was glad to find that I was wrong about this. The book is not a dry text, although there are plenty of definitions, theorems and proofs. What makes it come alive is the general understanding of financial instruments and their application. For a graduate student in mathematics this text should not represent too much of a challenge (assuming they have the required background); for practitioners in finance its accessibility will depend only on their mathematics background, as it is written at a sophisticated level.

In general, lots of patience and eagerness to learn the material will get you through a book of this size and content.

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.