Customer Reviews

Methods of Mathematical Finance

5 star:		(3)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

The application of highly sophisticated mathematical techniques to finance is now commonplace and is considered also of great practical importance. Mathematical modeling in finance is now very entrenched in investment houses and trading firms and this will only increase in years to come. This book is an excellent overview of mathematical finance and is written for mathematicians who have no background in finance. The book could be read easily by anyone with background in stochastic processes at the level of the author's earlier book "Brownian Motion and Stochastic Calculus". Since it is written for mathematicians, it follows a "definition-theorem-proof" format. However the authors do interject a lot of explanation into the dialog, especially that concerning finance.

Chapter 1 is an overview of a Brownian motion model of financial markets. Financial assets are considered to have prices evolving continuously in time and driven by Brownian motion. They do however g!ive references for models that assume discontinuous asset prices. The authors define a financial market rigorously in terms of (progressively) measurable processes for the risk-free rate, mean rate of return, dividend rate, and volatility. The after a discussion of portfolio, gains, income, and wealth processes, the authors define a notion of a viable market, namely one where there are no arbitrage opportunities. They then define standard and complete financial model and characterize their properties in terms of martingales.

Chapter 2 is a treatment of options pricing theory, with the assumption of a complete standard, financial market. These contingent claims are given a brief historical introduction at the beginning of the chapter. European contigent claims are treated first, followed by a discussion of forward and futures contracts. The Black-Scholes option pricing formula is then derived. American contingent claims are then discussed and defined as an income proc!ess and a settlement process. With the assumption that the discount payoff process is bounded from below and continuous, the value of the American contingent claim is given in terms of the Snell envelope of the payoff process. The discussion illustrates the difficulties in valuing American claims, based as they are on an arbitrary exercise time.

Chapter 3 is a study of a "small" single investor who begins with an initial endowment and invests in a standard complete market. The discussion reads more like one from a book on utility theory and portfolio analysis. Indeed, the Legendre transform of the utility function appears when attempting to mazimize utility from consumption plus expected utility from terminal wealth. The (nonlinear) Hamilton-Jacobi-Bellman equation appears in thes considerations as expected.

In chapter 4, the equilibrium problem is considered. In such a model, security prices are determined by the law of supply and demand. There are a finite !number of agents with utility functions and there are endowment processes. The endowments can be traded via a financial market of stocks and money market funds. The goal of the chapter is to find the equilibrium condition where endowments are consumed and the net supply of securities is zero. The authors give a rigorous proof of the existence and uniqueness of equilibrium. In addition, they give interesting examples of equilibrium markets that can be computed explicitly.

The next chapter is much more involved and studies how to do arbitrage pricing in incomplete markets. Portfolio constraints force the market to be incomplete, and the authors show how buyers and sellers in such a market can calculate the hedging price of a claim in terms of "dual" processes in a family of auxiliary markets. Since this is a constrained optimization problem, one would naturally think Lagrange multipliers would appear, and this is indeed the case, with the dual processes being the analog!ue of Lagrange multipliers. The usual unconstrained problem then is the result of this. Their approach here is extended in the last chapter of the book where the problem of optimal consumption and investment in a constrained financial market is considered. This is specialized to a deterministic case and the dual to the constrained problem satisfies a linear Hamilton-Jacobi-Bellman equation. This duality between the Lagrangian and Hamiltonian points of view is not surprising to the astute reader (and particularly the physicist reader).

For those working in higher levels of pure mathematics or physics Ioannis Karatzas's and Steven E. Shreve's Methods of Mathematical Finance will be the most accessible for helping you understand what all the fuss is about in finance and Wall Street. From the groves of academe, finance as it is practiced looks like so much "nonsense on stilts." However, serious intellectual work has been done examining finance and transactions under limits, spaces, stochastic paths and operators, and this work is the most rigorous explication of the foundations of this thinking, and its most natural extensions and applications.

This work is explicitly not for MBAs or other `phynance-lite" types who view interest rates as single factor driven and think the alpha and omega of option pricing as the Black-Scholes model. While the work rigorously addresses interest rates and option pricing from a mathematical standpoint, it is better thought of as applying Brownian motion to contingent events and time series, which for the purposes of this volume are financial values and the volatility of outcomes.

Another audience will be advanced students studying financial engineering or mathematical finance. This book is foundational required reading in most of the French DEA programs dealing with stochastic applications to finance.
One major caution: unless you have an intuitive grasp of programming from reading math presented in the "definition-theorem-proof" form of academia, you will be at a loss as to how to bridge this work to a practical application. I know of students who floundered around with Mathematica and this volume before coming across more accessible works written for practitioners and programmers in mind. This work is for those well trained in mathematics who want to learn about finance. For learning about programming optimal savings and consumption portfolios, option prices, etc. other works, such as those by Mark Joshi, are your better choice.

Very rigorous and methematically precise, but how can this text not even mention Ito's lemma? Well, because it isn't really a "sequel to Brownian Motion and Stochastic Calculus by the same authors" but more like the second half of that book. Unless you have their BM&SC or another similar reference by your side you won't get very far . . . and this fact is not at all apparent from reading the editorial description or jacket review.

For a self-containted text with both the basic math background AND the finance I recommend either Lamberton and Lapeyre (fairly complete but with some technical proofs referred to BM&SC) or Joshi (lots of applications, less mathy). Neither of these will be as comprehensive or rigorous as the 2-volume Karatzas and Shreve but both are good introductions to the subject.

This is the most mathematically rigorous treatment of asset pricing that is available. That's why this gets 4 stars. But this book is certainly not for everyone, which is why the 1 star is left out.

I am not kidding when I say this book is only for the most mathematically inclined. I have read many texts that require a decent amount of mathematics and I have already encountered the concepts that are discussed in this book, and I still found it quite a challenging read. In my opinion, you need to have a good grasp of probability and random processes to enjoy this book. For those who would like a lighter read, I suggest looking into Cochrane's Asset Pricing, LeRoy and Werner's Principles of Financial Economics, or even Skiadas' Asset Pricing Theory.

As a general matter, this book covers the major topics in asset pricing (e.g., financial markets, contingent claims, complete markets, consumption, investment, hedging, etc.) If you are technically inclined and you are up for the challenge, you should go ahead and read this book by all means. It is a very good book with little or no fluff.

The book is challenging. But if you want to do real good work in finance. You must read it.