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Methods of Mathematical Finance

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).

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