Table of Contents
Preface vii
About the Author ix
1 Introduction 1
1.1 Chapter Outline 4
1.2 Related Studies 8
1.3 Notation 12
Part I Drawdown Measures 15
2 Drawdowns Preceding Drawups in a Finite Time-Horizon 17
2.1 Random Walk Model 18
2.2 Brownian Motion with Drift Model 24
2.3 One-Dimensional Linear Diffusion Model 28
2.3.1 Analytical results 29
2.3.2 Example: Brownian motion with drift 34
2.4 Applications 35
2.5 Concluding Remarks 38
2.6 Proof of Lemmas 39
3 Drawdowns and the Speed of Market Crashes 41
3.1 Mathematical Formulation and Analytical Results 43
3.2 Progressive Enlargement of Filtrations by ga 44
3.3 Proof of the Main Results 46
3.4 Applications 52
3.4.1 Brownian motion with drift 53
3.4.2 The CEV model 54
3.5 Concluding Remarks 56
3.6 Proof of Lemmas 56
4 Frequency of Drawdowns in a Brownian Motion Model 59
4.1 Preliminaries 61
4.2 Sequences of Drawdown Times 63
4.2.1 Drawdown times with recovery 63
4.2.2 Drawdown times without recovery 66
4.3 Numerical Results 74
4.4 Insurance of Frequent Relative Drawdowns 74
4.5 Concluding Remarks 78
4.6 Proof of Lemmas 79
5 Occupation Times Related to Drawdowns 81
5.1 Definition of Occupation Times 83
5.2 Analytical Results 84
5.2.1 Occupation time below a level until the first exit time 84
5.2.2 Occupation time below a level until the first passage of drawdown 85
5.2.3 Occupation time of the drawdown process until the first passage of drawdown 86
5.2.4 Occupation time of the drawup process until the first passage of drawdown 89
5.2.5 Occupation time of the drawdown process at an independent exponential time 90
5.3 Examples 91
5.3.1 Brownian motion with drift 91
5.3.2 Three-dimensional Bessel process (BES(3)) 92
5.4 Applications 93
5.4.1 Probabilities regarding drawdowns and defaults 93
5.4.2 Option pricing for the drawdown process 94
5.5 Concluding Remarks 96
5.6 Proof of Lemmas 96
6 Duration of Drawdowns under Lévy Models 99
6.1 Preliminaries 101
6.1.1 Spectrally negative Lévy processes and scale functions 101
6.1.2 The ascending ladder process of general Lévy processes 104
6.2 Magnitude of Drawdowns Revisited 106
6.3 Asymptotics of Magnitude of Drawdowns 107
6.3.1 Spectrally negative Lévy processes 108
6.3.2 A class of Lévy models with two-sided jumps 110
6.4 Duration of Drawdowns 112
6.4.1 Bounded variation case 114
6.4.2 Unbounded variation case 118
6.5 Examples 122
6.6 Concluding Remarks 125
6.7 Proof of Lemmas and the Extended Continuity Theorem 125
Part II Applications of Drawdown 131
7 Maximum Drawdown Insurance Using Options 133
7.1 Setup and Replicating Instruments 136
7.2 Static Hedging of the K-drawdown Preceding a K-drawup with One-Touch Knockouts 139
7.3 Semi-static Hedging of the Maximum Drawdown with One-Touch Knockouts 141
7.4 Semi-static Replication with One-Touches 143
7.4.1 Hedging the maximum drawdown 145
7.4.2 Hedging the K-drawdown preceding a K-drawup 146
7.5 Semi-static Replication with Path-Independent Options 148
7.5.1 Hedging the maximum drawdown 150
7.5.2 Hedging the K-drawdown preceding a K-drawup 151
7.6 Static Hedging of the K-relative Drawdown Preceding a K-relative Drawup with One-Touch Knockouts 152
7.7 Semi-static Replication with One-Touches in Geometric Models 153
7.7.1 Hedging the maximum relative drawdown 155
7.7.2 Hedging the K-relative drawdown preceding a K-relative drawup 159
7.8 Semi-static Replication with Path-Independent Options in Geometric Models 162
7.8.1 Hedging the maximum relative drawdown 164
7.8.2 Hedging the K-relative drawdown preceding a K-relative drawup 165
7.9 Poisson Jump Processes 167
7.9.1 Arithmetic case 167
7.9.2 Geometric case 169
7.10 Concluding Remarks 172
7.11 Proof for GBM 172
8 Fair Premiums of Drawdown Insurances 175
8.1 Model for Drawdown Insurance 176
8.1.1 Drawdown insurance and fair premium 178
8.2 Cancellable Drawdown Insurance 180
8.2.1 Contract value decomposition 181
8.2.2 Optimal cancellation strategy 182
8.3 Incorporating Drawup Contingency 187
8.3.1 The finite maturity case 187
8.3.2 Perpetual case 189
8.4 Drawdown Insurance on a Defaultable Stock 191
8.5 Concluding Remarks 197
8.6 Proof of Lemmas 197
9 Optimal Trading with a Trailing Stop 203
9.1 Model Formulation 204
9.1.1 Standing assumption 207
9.2 Optimal Trading with a Fixed Stop-Loss 208
9.2.1 Optimal liquidation subject to a stop-loss exit 209
9.3 Optimal Trading with a Trailing Stop 210
9.3.1 Optimal liquidation 210
9.3.2 Optimal acquisition with a trailing stop 214
9.4 Case Study: Trading with a Trailing Stop under the Exponential OU Model 216
9.4.1 Value function and optimal strategy 217
9.4.2 Sensitivity analysis 220
9.5 Concluding Remarks 222
Appendix. Briefly on One-Dimensional Linear Diffusions 225
Bibliography 229
Index 239
