Methods of Mathematical Finance [Hardcover]

Ioannis Karatzas (Author), Steven E. Shreve (Author)

Book Description

ISBN-10: 0387948392 | ISBN-13: 978-0387948393 | Publication Date: August 13, 1998 | Edition: Corrected

This sequel to Brownian Motion and Stochastic Calculus by the same authors develops contingent claim pricing and optimal consumption/investment in both complete and incomplete markets, within the context of Brownian-motion-driven asset prices. The latter topic is extended to a study of equilibrium, providing conditions for existence and uniqueness of market prices which support trading by several heterogeneous agents. Although much of the incomplete-market material is available in research papers, these topics are treated for the first time in a unified manner. The book contains an extensive set of references and notes describing the field, including topics not treated in the book. This book will be of interest to researchers wishing to see advanced mathematics applied to finance. The material on optimal consumption and investment, leading to equilibrium, is addressed to the theoretical finance community. The chapters on contingent claim valuation present techniques of practical importance, especially for pricing exotic options.

Editorial Reviews

Review

"The book under review deals with the applications of stochastic analysis and optimal control theory to various problems arising in modern mathematical finance. In contrast to several other books on mathematical finance which appeared in recent years, this book deals not only with the so-called partial equilibrium approach (i.e., the arbitrage pricing of European and American contingent claims) but also with the general equilibrium approach (i.e., with the equilibrium specification of prices of primary assets). A major part of the book is devoted to solving valuation and portfolio optimization problems under market imperfections, such as market incompleteness and portfolio constraints. ... Undoubtedly, the book constitutes a valuable research-level text which should be consulted by anyone interested in the area. Unlike other currently available monographs, it provides an exhaustive and up-to-date treatment of portfolio optimization and valuation problems under constraints. It is also quite suitable as a textbook for an advanced course on mathematical finance."  (Marek RutKowski, Mathematical Reviews)

From the Publisher

Written by two of the best-known researchers in mathematical finance, this book will appeal to theorists and practitioners in this very active research area. It includes much material that has not appeared before in book-form. This book is a sequel to their GTM volume. The book closest in level is Musiela/Rutowski. According to Shreve, M/R is 100% applied while K/S has a mix of practical and theoretical resutls. M/R is more comprehensive. K/S concentrates on their research interests and includes more original work.

Product Details

• Hardcover: 422 pages
• Publisher: Springer; Corrected edition (August 13, 1998)
• Language: English
• ISBN-10: 0387948392
• ISBN-13: 978-0387948393
• Product Dimensions: 9.3 x 6.2 x 1.1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #1,254,630 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

42 of 48 people found the following review helpful:

5.0 out of 5 stars One of the best, June 30, 2001

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Methods of Mathematical Finance (Hardcover)

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).

8 of 11 people found the following review helpful:

5.0 out of 5 stars The acme of rigorous mathfin, not for the faint hearted, September 7, 2005

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Bachelier ""1004"" (Ile de France) - See all my reviews

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This review is from: Methods of Mathematical Finance (Hardcover)

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.

7 of 10 people found the following review helpful:

2.0 out of 5 stars NOT a stand-alone book, April 23, 2004

By 

J. Holman (San Francisco) - See all my reviews
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This review is from: Methods of Mathematical Finance (Hardcover)

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.