Hidden Markov Models in Finance (eBook)

Preface 6
Contents 10
List of Contributors 16
Biographical Notes 18
Editors 18
Contributors 19
1 Robustification of an On-line EM Algorithm for Modelling Asset Prices Within an HMM 24
1.1 Introduction 24
1.2 Hidden Markov Model Framework for Asset Returns 26
1.3 Essential Steps in Elliott's Algorithm 27
1.3.1 Change of Measure 27
1.3.2 Filtering for General Adapted Processes 27
1.3.3 Filter-Based EM Algorithm 30
1.3.4 Summary of Algorithm 31
1.4 Outliers in Asset Allocation Problem 31
1.4.1 Outliers in General 31
1.4.2 Time-Dependent Context: Exogenous and Endogenous Outliers 32
1.4.3 Evidence for Robustness Issue in Asset Allocation 33
1.5 Robust Statistics 35
1.5.1 Concepts of Robust Statistics 35
1.5.2 Our Robustification of the HMM: General Strategy 37
1.5.3 Robustification of Step (0) 38
1.5.4 Robustification of the E-Step 38
1.5.4.1 Crucial Optimality Theorem 39
1.5.4.2 Robustification of Steps (RN), (E) 39
1.5.5 Robustification of the (M)-Step 41
1.5.5.1 Shrinking Neighborhood Approach 41
1.5.5.2 Shrinking Neighborhood Approach with Weighted Observations 42
1.5.5.3 Robustification of Steps (M1) and (M2) 44
1.6 Implementation and Simulation 46
1.7 Conclusion 47
1.7.1 Contribution of This Paper 47
1.7.2 Outlook 49
Appendix 50
References 52
2 Stochastic Volatility or Stochastic Central Tendency: Evidence from a Hidden Markov Model of the Short-Term Interest Rate 55
2.1 Introduction 55
2.2 The Model 58
2.3 Maximum Likelihood Estimation 59
2.4 The Likelihood Function 59
2.5 The Interest Rate Model 61
2.6 Data 62
2.7 Results 67
2.8 Conclusion 73
References 74
3 An Econometric Model of the Term Structure of Interest Rates Under Regime-Switching Risk 76
3.1 Introduction 76
3.2 The Model 79
3.2.1 A Simple Representation of Markov Regime Shifts 79
3.2.2 Other State Variables 80
3.2.3 The Term Structure of Interest Rates 81
3.2.4 Bond Risk Premiums Under Regime Shifts 82
3.2.5 An Affine Regime-Switching Model 84
3.2.6 The Effects of Regime Shifts on the Yield Curve 87
3.3 Empirical Results 89
3.3.1 Data and Summary Statistics 89
3.3.2 Estimation Procedure 91
3.3.3 Discussions 95
3.4 Conclusion 101
References 101
4 The LIBOR Market Model: A Markov-Switching Jump Diffusion Extension 105
4.1 Introduction 105
4.2 Mathematical Preliminaries 107
4.3 The Log-Normal LIBOR Framework 110
4.3.1 An Introduction to the LIBOR Market Model: The Log-Normal Dynamics 111
4.3.2 Pricing of Caps and Floors in the Log-Normal LMM 113
4.4 The Markov-Switching Jump Diffusion (MSJD) Extension of the LMM 114
4.4.1 Presenting the Extended Framework 115
4.4.2 The Measure Changes and Its Consequences 117
4.4.2.1 The Measure Changes, the Wiener Process and the Compensator 117
4.4.2.2 The Measure Changes and the Markov Chain 119
4.5 Pricing in the MSJD Framework 120
4.5.1 Determining the Characteristic Function of YN-1 121
4.5.2 Determining the Characteristic Function of Yj, j=1,…,N-2 125
4.6 Calibration 125
4.6.1 The Data 126
4.6.2 Discussion of the Results of the Calibration 128
4.6.2.1 Most Likely Path and Infinitesimal Generator of the Markov Chain 128
4.6.2.2 Parameters Specifying the Compensator Measure with Respect to P 129
4.6.2.3 Volatility Parameters in the Model Specification Without Jumps 129
4.6.2.4 Final Estimates for Volatility and Jump Parameters 131
4.7 Conclusion 132
References 134
5 Exchange Rates and Net Portfolio Flows: A Markov-Switching Approach 137
5.1 Introduction 137
5.2 The Model 139
5.3 Data 140
5.4 Empirical Results 142
5.5 Conclusions 149
References 150
6 Hedging Costs for Variable Annuities Under Regime-Switching 153
6.1 Introduction 154
6.2 Hedging Costs 156
6.2.1 Derivation of the Pricing Equation 157
6.2.2 Events 159
6.2.2.1 Event Times 159
6.2.2.2 Withdrawal Strategy 159
6.2.2.3 Bonus 160
6.2.2.4 Withdrawal Not Exceeding the Contract Rate 160
6.2.2.5 Partial or Full Surrender 160
6.2.2.6 Ratchets 161
6.2.2.7 Simultaneous Events 161
6.2.3 Loss-Maximizing Strategies 161
6.2.4 Regime-Switching 162
6.3 Optimal Consumption 163
6.3.1 Utility PDE 164
6.3.2 Events 164
6.3.3 Consumption-Optimal Withdrawal 165
6.3.4 Regime-Switching 166
6.3.5 Hyperbolic Absolute Risk-Aversion 167
6.4 Numerical Method 167
6.4.1 Homogeneity 167
6.4.2 Localized Problem and Boundary Conditions 170
6.4.3 Determining the Hedging Cost Fee 171
6.5 Results 171
6.5.1 Loss-Maximizing and Contract Rate Withdrawal 171
6.5.1.1 Withdrawal Analysis 172
6.5.1.2 Management Rate 174
6.5.1.3 Alternate Fee Structure 174
6.5.2 Consumption-Optimal Withdrawal 174
6.5.2.1 Risk-Aversion 175
6.5.2.2 Taxation 177
6.6 Conclusion 179
Appendix 180
References 185
7 A Stochastic Approximation Approach for Trend-FollowingTrading 187
7.1 Introduction 187
7.2 Problem Formulation 188
7.3 Asymptotic Properties 191
7.4 Numerical Examples 196
References 203
8 A Hidden Markov-Modulated Jump Diffusion Model for EuropeanOption Pricing 205
8.1 Introduction 205
8.2 Hidden Regime-Switching Jump-Diffusion Market 208
8.3 Filtering Theory and Filtered Market 212
8.3.1 The Separation Principle 212
8.3.2 Filtering Equations 214
8.4 Generalized Esscher Transform in the Filtered Market 218
8.5 European-Style Option 223
8.6 Conclusion 227
References 227
9 An Exact Formula for Pricing American Exchange Options with Regime Switching 230
9.1 Introduction 230
9.2 Asset Price Dynamics 232
9.3 Problem Formulation 234
9.4 A Closed-Form Formula 238
9.5 Conclusion 243
References 243
10 Parameter Estimation in a Weak Hidden Markov Modelwith Independent Drift and Volatility 246
10.1 Introduction 246
10.2 Modelling Background 249
10.3 Filters and Parameter Estimation 250
10.4 Numerical Implementation 254
10.5 Conclusion 258
References 258
11 Parameter Estimation in a Regime-Switching Modelwith Non-normal Noise 260
11.1 Introduction 260
11.2 Model Set Up 261
11.3 Reference Probability Measure 262
11.4 Recursive Estimation 263
11.5 Parameter Estimation 265
11.5.1 EM Algorithm and the Estimation of Transition Probabilities 266
11.5.2 Student's t-Distributed Noise Term 267
11.6 Numerical Application of the Filters 271
11.6.1 Filtering Using Simulated Data 271
11.6.2 Application of the Filters to Observed Market Data 277
11.7 Conclusions 279
References 280