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E-raamat: Methods of Mathematical Finance

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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.

Arvustused

"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)

Preface vii
1 A Brownian Model of Financial Markets
1(35)
1.1 Stocks and a Money Market
1(5)
1.2 Portfolio and Gains Processes
6(4)
1.3 Income and Wealth Processes
10(1)
1.4 Arbitrage and Market Viability
11(5)
1.5 Standard Financial Markets
16(5)
1.6 Completeness of Financial Markets
21(6)
1.7 Financial Markets with an Infinite Planning Horizon
27(4)
1.8 Notes
31(5)
2 Contingent Claim Valuation in a Complete Market
36(52)
2.1 Introduction
36(3)
2.2 European Contingent Claims
39(4)
2.3 Forward and Futures Contracts
43(4)
2.4 European Options in a Constant-Coefficient Market
47(7)
2.5 American Contingent Claims
54(6)
2.6 The American Call Option
60(7)
2.7 The American Put Option
67(13)
2.8 Notes
80(8)
3 Single-Agent Consumption and Investment
88(71)
3.1 Introduction
88(2)
3.2 The Financial Market
90(1)
3.3 Consumption and Portfolio Processes
91(3)
3.4 Utility Functions
94(3)
3.5 The Optimization Problems
97(4)
3.6 Utility from Consumption and Terminal Wealth
101(10)
3.7 Utility from Consumption or Terminal Wealth
111(7)
3.8 Deterministic Coefficients
118(18)
3.9 Consumption and Investment on an Infinite Horizon
136(14)
3.10 Maximization of the Growth Rate of Wealth
150(3)
3.11 Notes
153(6)
4 Equilibrium in a Complete Market
159(40)
4.1 Introduction
159(2)
4.2 Agents, Endowments, and Utility Functions
161(2)
4.3 The Financial Market: Consumption and Portfolio Processes
163(4)
4.4 The Individual Optimization Problems
167(3)
4.5 Equilibrium and the Representative Agent
170(8)
4.6 Existence and Uniqueness of Equilibrium
178(11)
4.7 Examples
189(7)
4.8 Notes
196(3)
5 Contingent Claims in Incomplete Markets
199(61)
5.1 Introduction
199(2)
5.2 The Model
201(3)
5.3 Upper Hedging Price
204(1)
5.4 Convex Sets and Support Functions
205(3)
5.5 A Family of Auxiliary Markets
208(3)
5.6 The Main Hedging Result
211(9)
5.7 Upper Hedging with Constant Coefficients
220(5)
5.8 Optimal Dual Processes
225(13)
5.9 Lower Hedging Price
238(16)
5.10 Lower Hedging with Constant Coefficients
254(3)
5.11 Notes
257(3)
6 Constrained Consumption and Investment
260(63)
6.1 Introduction
260(1)
6.2 Utility Maximization with Constraints
261(5)
6.3 A Family of Unconstrained Problems
266(9)
6.4 Equivalent Optimality Conditions
275(9)
6.5 Duality and Existence
284(7)
6.6 Deterministic Coefficients, Cone Constraints
291(11)
6.7 Incomplete Markets
302(8)
6.8 Higher Interest Rate for Borrowing Than for Investing
310(8)
6.9 Notes
318(5)
Appendix A. Essential Supremum of a Family of Random Variables 323(4)
Appendix B. On the Model of Section 1.1 327(8)
Appendix C. On Theorem 6.4.1 335(14)
Appendix D. Optimal Stopping for Continuous-Parameter Processes 349(14)
Appendix E. The Clark Formula 363(8)
References 371(32)
Index 403


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