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AmericanStyle Derivatives-Valuation and Computation.pdf
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American-Style Derivatives Valuation and Computation
Jérôme Detemple
Published in 2006 by
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487–2742
© 2006 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group
International Standard Book Number-10:1-58488-567-X (Hardcover)
International Standard Book Number-13:978-1-58488-567-2 (Hardcover)
Library of Congress Card Number 2005052861
1 Introduction 1
2 European Contingent Claims 7
2.1 Definitions 7
2.2 The Economy 8
2.3 Attainable Contingent Claims 10
2.4 Valuation of Attainable Claims 14
2.5 Claims Involving Negative Payoffs 16
2.6 The Structure of Contingent Claims’ Prices 18
2.7 Changes of Numeraire and Valuation 19
2.8 Option and Forward Contracts 21
2.9 Markets with Deterministic Coefficients 24
2.10 Markets with Multiple Assets 28
2.11 Appendix: Proofs 29
3 American Contingent Claims 37
3.1 Contingent Claims with Random Maturity 37
3.2 American Contingent Claims 39
3.3 Exercise Premium Representations 41
3.4 A Duality Formula: Upper Price Bounds 44
3.5 American Options and Forward Contracts 45
3.6 Multiple Underlying Assets 47
3.7 Appendix: Proofs 48
4 Standard American Options 55
4.1 The Immediate Exercise Region 55
4.2 The Call Price Function 57
4.3 Early Exercise Premium Representation 59
4.4 A One-Dimensional Integral Equation 61
4.5 Hedging 62
4.6 Diffusion Processes 63
4.7 Floating Strike Asian Options 67
4.8 American Forward Contracts 69
4.9 Appendix: Proofs 71
Contents
© 2006 by Taylor & Francis Group, LLC
CONTENTS
5 Barrier and Capped Options 85
5.1 Barrier Options 85
5.1.1 Definitions and Literature 85
5.1.2 Valuation 86
5.2 Capped Options 89
5.2.1 Definitions, Examples and Literature 89
5.2.2 Constant Cap 89
5.2.3 Capped Options with Growing Caps 93
5.2.4 Stochastic Cap, Interest Rate and Volatility 96
5.3 Diffusion Processes 97
5.4 Appendix: Proofs 98
6 Options on Multiple Assets 107
6.1 Definitions, Examples and Literature 107
6.2 The Financial Market 109
6.3 Call Options on the Maximum of 2 Prices 109
6.3.1 Exercise Region of a Max-Call Option 110
6.3.2 Valuation of Max-Call Options 114
6.3.3 Dual Strike Max-Options 119
6.3.4 Put Options on the Minimum of 2 Prices 120
6.3.5 Economic Implications 120
6.4 American Spread Options 121
6.4.1 Exercise Region and Valuation 121
6.4.2 Options to Exchange One Asset for Another 123
6.4.3 Exchange Options with Proportional Caps 124
6.5 Options on an Average of 2 Prices 125
6.5.1 Geometric Averaging 125
6.5.2 Arithmetic Averaging 126
6.6 Call Options on the Minimum of 2 Prices 129
6.6.1 Exercise Region of a Min-Call Option 129
6.6.2 The EEP Representation 132
6.6.3 Integral Equations for the Boundary Components 135
6.7 Appendix A: Derivatives on Multiple Assets 135
6.8 Appendix B: Proofs 143
7 Occupation Time Derivatives 155
7.1 Background and Literature 155
7.2 Definitions 156
7.3 Symmetry Properties 157
7.4 Quantile Options 158
7.4.1 Contractual Specification 158
7.4.2 The Distribution of an α-Quantile 159
7.4.3 Pricing Quantile Options 159
7.4.4 A Reduction in Dimensionality 162
7.4.5 Quantile Contingent Claims 163
7.5 Parisian Options 163
© 2006 by Taylor & Francis Group, LLC
CONTENTS
7.5.1 Contractual Specification 164
7.5.2 Parity and Symmetry Relations 165
7.5.3 Pricing Parisian Options 166
7.5.4 Parisian Contingent Claims 169
7.6 Cumulative Parisian Contingent Claims 170
7.6.1 Definitions and Parity/Symmetry Relations 170
7.6.2 Pricing Cumulative Barrier Claims 171
7.6.3 Standard and Exotic Cumulative Barrier Options 172
7.7 Step Options 173
7.7.1 Contractual Specification 173
7.7.2 Pricing European-style Step Options 174
7.8 American Occupation Time Derivatives 175
7.8.1 Early Exercise Premium Representation 175
7.8.2 Valuation in the Standard Model 176
7.9 Multiasset Claims 179
7.9.1 Symmetry Properties 179
7.9.2 Valuation 181
7.10 Appendix: Proofs 181
8 Numerical Methods 195
8.1 Numerical Methods for American Options 195
8.2 Integral Equation Methods 197
8.3 Exercise Time Approximations: LBA-LUBA 199
8.3.1 A Lower Bound for the Option Price 199
8.3.2 A Lower Bound for the Exercise Boundary 200
8.3.3 An Upper Bound for the Option Price 201
8.3.4 Price Approximations 202
8.4 Diffusion Processes 202
8.4.1 Integral Equation Methods 203
8.4.2 Stopping Time Approximations: LBA and LUBA 203
8.5 Other Recent Approaches 204
8.5.1 Lattice Methods: Binomial Black-Scholes Algorithm 204
8.5.2 Integral Equation: Non-linear Approximations 204
8.5.3 Monte Carlo Simulation 205
8.6 Performance Evaluation 205
8.6.1 Experiment Design 206
8.6.2 Results and Discussion 206
8.7 Methods for Multiasset Options 207
8.7.1 Lattice Methods 207
8.7.2 Monte Carlo Simulation 209
8.7.3 Monte Carlo Simulation and Duality 211
8.8 Methods for Occupation Time Derivatives 212
8.8.1 Laplace Transforms 212
8.8.2 PDE-based Methods 212
8.8.3 Binomial/Trinomial Lattices 213
8.9 Appendix: Proofs 215
© 2006 by Taylor & Francis Group, LLC
CONTENTS
Bibliography 217
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