Stochastic financial models

Title:

Personal Author:

Kennedy, Douglas.

Series:

Chapman & Hall/CRC financial mathematics series

Chapman & Hall/CRC financial mathematics series.

Publication Information:

Boca Raton, FL : Chapman & Hall/CRC, 2010.

Physical Description:

257 p. : ill. ; 25 cm.

ISBN:

9781420093452

Subject Term:

Investments -- Mathematical models.

Stochastic analysis.

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010237344	HG4515.2 K46 2010	Open Access Book	Book	Searching... Unknown

Filling the void between surveys of the field with relatively light mathematical content and books with a rigorous, formal approach to stochastic integration and probabilistic ideas, Stochastic Financial Models provides a sound introduction to mathematical finance. The author takes a classical applied mathematical approach, focusing on calculations rather than seeking the greatest generality.

Developed from the esteemed author's advanced undergraduate and graduate courses at the University of Cambridge, the text begins with the classical topics of utility and the mean-variance approach to portfolio choice. The remainder of the book deals with derivative pricing. The author fully explains the binomial model since it is central to understanding the pricing of derivatives by self-financing hedging portfolios. He then discusses the general discrete-time model, Brownian motion and the Black-Scholes model. The book concludes with a look at various interest-rate models. Concepts from measure-theoretic probability and solutions to the end-of-chapter exercises are provided in the appendices.

By exploring the important and exciting application area of mathematical finance, this text encourages students to learn more about probability, martingales and stochastic integration. It shows how mathematical concepts, such as the Black-Scholes and Gaussian random-field models, are used in financial situations.

Author Notes

Douglas Kennedy is a Fellow of Trinity College in Cambridge, UK.

Reviews 1

Choice Review

The modeling of financial processes provides an interesting and timely application of topics such as probability, martingales, and stochastic integration, which can be studied on their own as purely mathematical subjects. Kennedy (Trinity College, UK) intends to fill a gap between works that have somewhat lighter mathematical content and prerequisites (e.g., Sheldon Ross's An Elementary Introduction to Mathematical Finance, 2nd ed., 2002), and those that he describes as taking no prisoners with respect to their formal approach and mathematical rigor. But even so, the prerequisites here, from the standpoint of a standard US undergraduate mathematics curriculum, are highly nontrivial. At the very least, students must have completed a serious course in mathematical probability, preferably with a measure-theoretic component, and ideally with some stochastic integration. The work's six chapters are titled "Portfolio Choice," "The Binomial Model," "A General Discrete-Time Model," "Brownian Motion, "The Black-Scholes Model," and "Interest-Rate Models." An appendix provides exercise solutions. Summing Up: Recommended. Very advanced upper-division undergraduates and graduate students. D. Robbins Trinity College (CT)

Preface	p. ix
1 Portfolio Choice	p. 1
1.1 Introduction	p. 1
1.2 Utility	p. 2
1.2.1 Preferences and utility	p. 2
1.2.2 Utility and risk aversion	p. 7
1.3 Mean-variance analysis	p. 9
1.3.1 Introduction	p. 9
1.3.2 All risky assets	p. 9
1.3.3 A riskless asset	p. 14
1.3.4 Mean-variance analysis and expected utility	p. 18
1.3.5 Equilibrium: the capital-asset pricing model	p. 19
1.4 Exercises	p. 20
2 The Binomial Model	p. 25
2.1 One-period model	p. 25
2.1.1 Introduction	p. 25
2.1.2 Hedging	p. 26
2.1.3 Arbitrage	p. 28
2.1.4 Utility maximization	p. 29
2.2 Multi-period model	p. 31
2.2.1 Introduction	p. 31
2.2.2 Dynamic hedging	p. 33
2.2.3 Change of probability	p. 40
2.2.4 Utility maximization	p. 42
2.2.5 Path-dependent claims	p. 44
2.2.6 American claims	p. 49
2.2.7 The non-standard multi-period model	p. 54
2.3 Exercises	p. 59
3 A General Discrete-Time Model	p. 63
3.1 One-period model	p. 63
3.1.1 Introduction	p. 63
3.1.2 Arbitrage	p. 69
3.2 Multi-period model	p. 73
3.2.1 Introduction	p. 73
3.2.2 Pricing claims	p. 76
3.3 Exercises	p. 81
4 Brownian Motion	p. 83
4.1 Introduction	p. 83
4.2 Hitting-time distributions	p. 85
4.2.1 The reflection principle	p. 85
4.2.2 Transformations of Brownian motion	p. 93
4.2.3 Computations using martingales	p. 94
4.3 Girsanov's Theorem	p. 97
4.4 Brownian motion as a limit	p. 100
4.5 Stochastic calculus	p. 102
4.6 Exercises	p. 109
5 The Black-Scholes Model	p. 113
5.1 Introduction	p. 113
5.2 The Black-Scholes formula	p. 114
5.2.1 Derivation	p. 114
5.2.2 Dependence on the parameters: the Greeks	p. 116
5.2.3 Volatility	p. 119
5.3 Hedging and the Black-Scholes equation	p. 123
5.3.1 Self-financing portfolios	p. 123
5.3.2 Dividend-paying claims	p. 128
5.3.3 General terminal-value claims	p. 130
5.3.4 Specific terminal-value claims	p. 134
5.3.5 Utility maximization	p. 137
5.3.6 American claims	p. 143
5.4 Path-dependent claims	p. 146
5.4.1 Forward-start and lookback options	p. 146
5.4.2 Barrier options	p. 150
5.5 Dividend-paying assets	p. 156
5.6 Exercises	p. 159
6 Interest-Rate Models	p. 165
6.1 Introduction	p. 165
6.2 Survey of interest-rate models	p. 168
6.2.1 One-factor models	p. 168
6.2.2 Forward-rate and market models	p. 172
6.3 Gaussian random-field model	p. 174
6.3.1 Introduction	p. 174
6.3.2 Pricing a caplet on forward rates	p. 178
6.3.3 Markov properties	p. 182
6.3.4 Finite-factor models and restricted information	p. 188
6.4 Exercises	p. 190
A Mathematical Preliminaries	p. 193
A.1 Probability background	p. 193
A.1.1 Probability spaces	p. 193
A.1.2 Conditional expectations	p. 194
A.1.3 Change of probability	p. 194
A.1.4 Essential supremum	p. 196
A.2 Martingales	p. 196
A.3 Gaussian random variables	p. 198
A.3.1 Univariate normal distributions	p. 198
A.3.2 Multivariate normal distributions	p. 200
A.4 Convexity	p. 204
B Solutions to the Exercises	p. 207
B.1 Portfolio Choice	p. 207
B.2 The Binomial Model	p. 213
B.3 A General Discrete-Time Model	p. 221
B.4 Brownian Motion	p. 226
B.5 The Black-Scholes Model	p. 231
B.6 Interest-Rate Models	p. 240
Further Reading	p. 247
References	p. 249
Index	p. 253

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