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Brownian Motion Calculus

by: Wiersema, Ubbo F.

Brownian Motion Calculus

by: Wiersema, Ubbo F.

ISBN 13:
9780470021705
ISBN 10:
0470021705
Edition: 1st
Format: Paperback
Copyright: 12/08/2008
Publisher: WILEY

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Summary

Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website.

Preface	p. xiii
Brownian Motion	p. 1
Origins	p. 1
Brownian Motion Specification	p. 2
Use of Brownian Motion in Stock Price Dynamics	p. 4
Construction of Brownian Motion from a Symmetric Random Walk	p. 6
Covariance of Brownian Motion	p. 12
Correlated Brownian Motions	p. 14
Successive Brownian Motion Increments	p. 16
Numerical Illustration	p. 17
Features of a Brownian Motion Path	p. 19
Simulation of Brownian Motion Paths	p. 19
Slope of Path	p. 20
Non-Differentiability of Brownian Motion Path	p. 21
Measuring Variability	p. 24
Exercises	p. 26
Summary	p. 29
Martingales	p. 31
Simple Example	p. 31
Filtration	p. 32
Conditional Expectation	p. 33
General Properties	p. 34
Martingale Description	p. 36
Martingale Construction by Conditioning	p. 36
Martingale Analysis Steps	p. 37
Examples of Martingale Analysis	p. 37
Sum of Independent Trials	p. 37
Square of Sum of Independent Trials	p. 38
Product of Independent Identical Trials	p. 39
Random Process B(t)	p. 39
Random Process exp[B(t) - 1/2t]	p. 40
Frequently Used Expressions	p. 40
Process of Independent Increments	p. 41
Exercises	p. 42
Summary	p. 42
Itō Stochastic Integral	p. 45
How a Stochastic Integral Arises	p. 45
Stochastic Integral for Non-Random Step-Functions	p. 47
Stochastic Integral for Non-Anticipating Random Step-Functions	p. 49
Extension to Non-Anticipating General Random Integrands	p. 52
Properties of an Itō Stochastic Integral	p. 57
Significance of Integrand Position	p. 59
Itō integral of Non-Random Integrand	p. 61
Area under a Brownian Motion Path	p. 62
Exercises	p. 64
Summary	p. 67
A Tribute to Kiyosi Itō	p. 68
Acknowledgment	p. 72
Itō Calculus	p. 73
Stochastic Differential Notation	p. 73
Taylor Expansion in Ordinary Calculus	p. 74
Itō's Formula as a Set of Rules	p. 75
Illustrations of Itō's Formula	p. 78
Frequent Expressions for Functions of Two Processes	p. 78
Function of Brownian Motion f[B(t)] 80
Function of Time and Brownian Motion f[t, B(t)]	p. 82
Finding an Expression for $^Tt=0 B(t) dB(t)	p. 83
Change of Numeraire	p. 84
Deriving an Expectation via an ODE	p. 85
Lévy Characterization of Brownian Motion	p. 87
Combinations of Brownian Motions	p. 89
Multiple Correlated Brownian Motions	p. 92
Area under a Brownian Motion Path - Revisited	p. 95
Justification of Itō's Formula	p. 96
Exercises	p. 100
Summary	p. 101
Stochastic Differential Equations	p. 103
Structure of a Stochastic Differential Equation	p. 103
Arithmetic Brownian Motion SDE	p. 104
Geometric Brownian Motion SDE	p. 105
Ornstein-Uhlenbeck SDE	p. 108
Mean-Reversion SDE	p. 110
Mean-Reversion with Square-Root Diffusion SDE	p. 112
Expected Value of Square-Root Diffusion Process	p. 112
Coupled SDEs	p. 114
Checking the Solution of a SDE	p. 115
General Solution Methods for Linear SDEs	p. 115
Martingale Representation	p. 120
Exercises	p. 123
Summary	p. 124
Option Valuation	p. 127
Partial Differential Equation Method	p. 128
Martingale Method in One-Period Binomial Framework	p. 130
Martingale Method in Continuous-Time Framework	p. 135
Overview of Risk-Neutral Method	p. 138
Martingale Method Valuation of Some European Options	p. 139
Digital Call	p. 139
Asset-or-Nothing Call	p. 141
Standard European Call	p. 142
Links between Methods	p. 144
Feynman-Kač Link between PDE Method and Martingale Method	p. 144
Multi-Period Binomial Link to Continuous	p. 146
Exercise	p. 147
Summary	p. 148
Change of Probability	p. 151
Change of Discrete Probability Mass	p. 151
Change of Normal Density	p. 153
Change of Brownian Motion	p. 154
Girsanov Transformation	p. 155
Use in Stock Price Dynamics - Revisited	p. 160
General Drift Change	p. 162
Use in Importance Sampling	p. 163
Use in Deriving Conditional Expectations	p. 167
Concept of Change of Probability	p. 172
Relationship between Expected Values under Equivalent Probabilities	p. 174
Exercises	p. 174
Summary	p. 176
Numeraire	p. 179
Change of Numeraire	p. 179
In Discrete Time	p. 179
In Continuous Time	p. 182
Forward Price Dynamics	p. 184
Dynamics of Forward Price of a Bond	p. 184
Dynamics of Forward Price of any Traded Asset	p. 185
Option Valuation under most Suitable Numeraire	p. 187
Exchange Option	p. 187
Option on Bond	p. 188
European Call under Stochastic Interest Rate	p. 188
Relating Change of Numeraire to Change of Probability	p. 190
Change of Numeraire for Geometric Brownian Motion	p. 192
Change of Numeraire in LIBOR Market Model	p. 194
Application in Credit Risk Modelling	p. 198
Exercises	p. 200
Summary	p. 201
Annexes
Annex A: Computations with Brownian Motion	p. 205
Moment Generating Function and Moments of Brownian Motion	p. 205
Probability of Brownian Motion Position	p. 208
Brownian Motion Reflected at the Origin	p. 208
First Passage of a Barrier	p. 214
Alternative Brownian Motion Specification	p. 216
Annex B: Ordinary Integration	p. 221
Riemann Integral	p. 221
Riemann-Stieltjes Integral	p. 226
Other Useful Properties	p. 231
References	p. 234
Annex C: Brownian Motion Variability	p. 235
Quadratic Variation	p. 235
First Variation	p. 238
Annex D: Norms	p. 239
Distance between Points	p. 239
Norm of a Function	p. 242
Norm of a Random Variable	p. 244
Norm of a Random Process	p. 244
Reference	p. 246
Annex E: Convergence Concepts	p. 247
Central Limit Theorem	p. 247
Mean-Square Convergence	p. 248
Almost Sure Convergence	p. 249
Convergence in Probability	p. 250
Summary	p. 250
Answers to Exercises	p. 253
References	p. 299
Index	p. 303
Table of Contents provided by Ingram. All Rights Reserved.

Amazon no longer offers textbook rentals. We do!

Brownian Motion Calculus

Brownian Motion Calculus

9780470021705

0470021705

Summary

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Table of Contents

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