Methods of Mathematical Finance

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Springer Science & Business Media, Aug 13, 1998 - Business & Economics - 407 pages
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This book is intended for readers who are quite familiar with probability and stochastic processes but know little or nothing about ?nance. It is written in the de?nition/theorem/proof style of modern mathematics and attempts to explain as much of the ?nance motivation and terminology as possible. A mathematical monograph on ?nance can be written today only - cause of two revolutions that have taken place on Wall Street in the latter half of the twentieth century. Both these revolutions began at universities, albeit in economics departments and business schools, not in departments of mathematicsor statistics. Theyhaveledinexorably,however,to anes- lation in the level of mathematics (including probability, statistics, partial di?erential equations and their numerical analysis) used in ?nance, to a point where genuine research problems in the former ?elds are now deeply intertwined with the theory and practice of the latter. The ?rst revolution in ?nance began with the 1952 publication of “Po- folio Selection,” an early version of the doctoral dissertation of Harry Markowitz. This publication began a shift away from the concept of t- ing to identify the “best” stock for an investor, and towards the concept of trying to understand and quantify the trade-o?s between risk and - turn inherent in an entire portfolio of stocks. The vehicle for this so-called mean–variance analysis of portfolios is linear regression; once this analysis is complete, one can then address the optimization problem of choosing the portfolio with the largest mean return, subject to keeping the risk (i. e.
  

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Contents

A Brownian Model of Financial Markets
1
12 Portfolio and Gains Processes
6
13 Income and Wealth Processes
10
14 Arbitrage and Market Viability
11
15 Standard Financial Markets
16
16 Completeness of Financial Markets
21
17 Financial Markets with an Infinite Planning Horizon
27
18 Notes
31
46 Existence and Uniqueness of Equilibrium
178
47 Examples
189
48 Notes
196
Contingent Claims in Incomplete Markets
199
52 The Model
201
53 Upper Hedging Price
204
54 Convex Sets and Support Functions
205
55 A Family of Auxiliary Markets
208

Contingent Claim Valuation in a Complete Market
36
22 European Contingent Claims
39
23 Forward and Futures Contracts
43
24 European Options in a ConstantCoefficient Market
47
25 American Contingent Claims
54
26 The American Call Option
60
27 The American Put Option
67
28 Notes
80
SingleAgent Consumption and Investment
88
32 The Financial Market
90
33 Consumption and Portfolio Processes
91
34 Utility Functions
94
35 The Optimization Problems
97
36 Utility from Consumption and Terminal Wealth
101
37 Utility from Consumption or Terminal Wealth
111
38 Deterministic Coefficients
118
39 Consumption and Investment on an Infinite Horizon
136
310 Maximization of the Growth Rate of Wealth
150
311 Notes
153
Equilibrium in a Complete Market
159
42 Agents Endowments and Utility Functions
161
Consumption and Portfolio Processes
163
44 The Individual Optimization Problems
167
45 Equilibrium and the Representative Agent
170
56 The Main Hedging Result
211
57 Upper Hedging with Constant Coefficients
220
58 Optimal Dual Processes
225
59 Lower Hedging Price
238
510 Lower Hedging with Constant Coefficients
254
511 Notes
257
Constrained Consumption and Investment
260
62 Utility Maximization with Constraints
261
63 A Family of Unconstrained Problems
266
64 Equivalent Optimality Conditions
275
65 Duality and Existence
284
66 Deterministic Coefficients Cone Constraints
291
67 Incomplete Markets
302
68 Higher Interest Rate for Borrowing Than for Investing
310
69 Notes
318
Essential Supremum of a Family of Random Variables
323
On the Model of Section 11
326
On Theorem 641
335
Optimal Stopping for ContinuousParameter Processes
349
The Clark Formula
363
References
370
Symbol Index
403
Index
411
Copyright

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About the author (1998)

Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.