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Methods of mathematical finance. (English) Zbl 0941.91032
Applications of Mathematics 39. Berlin: Springer (ISBN 0-387-94839-2). xv, 407 p. (1998).
This mathematical monograph on finance models is intended to provide a thorough analysis for some of the financial instruments that are currently used to enhance the efficiency of an economy. Centred around the main ideas of the mean-variance analysis of portfolios, and of the derivative securities on the financial market the exposed models are built using stochastic calculus, are fit to the data by careful statistical estimation procedures, and require accurate and fast real-time numerical analysis instruments.
The book of loannis Karatzas and Steven E. Shreve is written in the definition / theorem / proof style of modern mathematics and attempts to explain in detail the finance motivation and terminology. An extensive bibliography and explanatory notes are included at the end of each chapter, trying to clear further some of the chapter contents, and to point the reader toward some of the topics not touched.
The six chapters of the book can be briefly summarized as follows: Chapter 1 (A Brownian model of financial markets) sets up the generally accepted Brownian-motion-driven model for financial markets. Because the coefficient processes in this model are themselves stochastic processes, this is one of the most general continuous-time model conceivable among those in which prices move continuously. There are introduced notions and results about portfolio and consumption rules, arbitrage, equivalent martingale measures, attainability of contingent claims, and the important concept of financial market completeness. Chapter 2 (Contingent claim valuation in a complete market) of the book lays out the theory of pricing and hedging contingent claims (i.e. derivative, securities) in the context of a complete market. The authors analyzed in detail the pricing and hedging for a number of different options, and include a section on “future” contract, derivation securities whose value is defined recursively.
Chapter 3 (Single-agent consumption and investment) considers the problem of a single agent faced with optimal consumption and investment decisions in the complete version of the market model exposed in the first chapter. Mathematical tools from stochastic calculus and partial differential equations of parabolic type permit a general treatment of the associated optimization problem. This theory is related to the Markowitz’s mean-variance analysis and provides a better understanding on how the financial markets operate.
Chapter 4 (Equilibrium in a complete market) contains results that naturally continue the premises within Chapter 3. Through the law of supply and demand collective actions of several individuals may determine the so-called equilibrium prices of securities in the market. Characterization of this equilibrium allows the study of questions about the effect of interventions in the market.
Chapter 5 (Contingent claims in incomplete markets) focuses to the more difficult issue of pricing and hedging contingent claims in markets with incompleteness or other constraints on individual investor’s portfolio choices. The approach based on “fictitious completion” for such a market, coupled with notions and results from convex analysis and duality theory, permits quite a general solution to the hedging problem.
The final Chapter 6 (Constrained consumption and investment) is using the approach developed in Chapter 5 in order to treat the optimal consumption / investment problem for incomplete or constrained financial markets, and for markets with different interest rates for borrowing and investing.
Five technical appendices investigate and complete with new aspects the mathematical and financial arguments for some of the important problem considered within the book chapters.

MSC:
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91Gxx Actuarial science and mathematical finance
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