Stochastic Filtering with Applications in Finance Book

Preface vii

1 Introduction: Stochastic Filtering in Finance

1.1 Filtering Problem 2

1.2 Examples of Filtering Applications 2

1.3 Linear Kalman Filter 3

1.4 Extended Kalman Filter (EKF) 6

1.5 Applying EKF to Interest Rate Model 7

1.6 Unscented Kalman Filter (UKF) for Nonlinear Models 10

1.7 Background to Particle Filter for Non Gaussian Problems 13

1.8 Particle Filter Algorithm 14

1.9 Unobserved Component Models 16

1.10 Concluding Remarks 19

2 Foreign Exchange Market - Filtering Applications

2.1 Mean Reversion in Real Exchange Rates 21

2.2 Common and Specific Components in Currency Movements 25

2.3 Persistent in Real Interest Rate Differentials 30

2.4 Risk Premia in Forward Exchange Rate 34

2.4.1 Approach based on Market Price of Risk (BCP) 36

2.4.2 Method of Wolff/Cheung 40

2.4.3 Data and Empirical Results 41

2.4.4 Summary of Section 2.4 43

2.5 Concluding Remarks 47

3 Equity Market - Filtering Applications

3.1 Introduction to Equity Price of Risk 49

3.1.1 A Model for Equity Price of Risk 51

3.1.2 Data Used for Empirical Study 52

3.1.3 Discussion of Empirical Results 53

3.1.4 Summary of Results 61

3.2 Economic Convergence in a Filtering Framework 62

3.2.1 Defining Convergence 64

3.2.2 Testing for Convergence 65

3.2.3 Testing Convergence - Dickey-Fuller 66

3.2.4 Testing Convergence - Kalman Filter 67

3.3 Ex-Ante Equity Risk Premium 69

3.3.1 Background to Ex Ante Risk Premium 69

3.3.2 A Model for Ex Ante Risk Premium 70

3.3.3 Filtering Ex Ante Risk Premium 72

3.3.4 Ex-Ante Risk Premium for UK 73

3.3.5 Summarizing Ex-Ante Risk Premium for UK 73

3.4 Concluding Remarks 75

4 Filtering Application-Inflation and the Macroeconomy

4.1 Background and Macroeconomic Issues 77

4.2 Inflation Targeting Countries and Data Requirement 79

4.3 Model for Inflation Uncertainties 80

4.4 Testing Fisher Hypothesis 82

4.5 Empirical Results and Analysis 83

4.6 Concluding Remarks 85

5 Interest Rate Model and Non-Linear Filtering

5.1 Background to HJM Model and the Related Literature 95

5.2 The Basic HJM Structure 97

5.3 Forward Rate Volatility: Deterministic Function of Time 100

5.4 Forward Rate Volatility: Stochastic 102

5.5 Estimation via Kalman Filtering 107

5.6 Preference-Free Approach to Bond Pricing 109

5.7 Concluding Remarks 112

Appendix 5.1 Arbitrage-Free SDE for the Bond Price 114

Appendix 5.2 Proof of Proposition 1 117

Appendix 5.3 Proof of Proposition 2 119

Appendix 5.4 Proof Proposition 3 122

6 Filtering and Hedging using Interest Rate Futures

6.1 Background Details 126

6.2 The Futures Price Model in the HJM Framework 127

6.3 Non-Linear Filter for Futures Price System 131

6.4 Data Used in Empirical Study 134

6.5 Empirical Results 135

6.6 Concluding Remarks 138

Appendix 6.1 139

7 A Multifactor Model of Credit Spreads

7.1 Background and Related Research 150

7.2 Variables Influencing Changes in Credit Spreads 151

7.3 Credit Spread and Default Risk 153

7.4 Credit Spread and Liquidity 155

7.5 Alternative Approach to Analyzing Credit Spread 156

7.6 Data Used 159

7.7 Multifactor Model for Credit Spread 160

7.8 Fitting the Model 162

7.9 Results 162

7.9.1 Results for Apr-96 to Mar-03 162

7.9.2 Results for Apr-96 to Mar-08 165

7.9.3 Model Performance 168

7.9.4 Discussion 168

7.10 Concluding Remarks 169

8 Credit Default Swaps - Filtering the Components

8.1 Background to Credit Default Swaps 185

8.2 What is in the Literature Already" 188

8.3 Credit Derivatives Market and iTraxx Indices 190

8.4 CDS Index Data and Preliminary Analysis 192

8.5 Focusing on Explanatory Variables 195

8.6 Methodology for Component Structure 201

8.6.1 Latent-Component Model for iTraxx Indices 201

8.6.2 State Space Model and Stochastic Filtering 203

8.6.3 Linear Regression Model for the Determinants of the CDS Components 204

8.7 Analyzing Empirical Results 205

8.7.1 Model Parameters and the Extracted Components 205

8.7.2 Determinants of the Extracted Components 207

8.8 Concluding Summary 211

9 CDS Options, Implied Volatility and Unscented Kalman Filter

9.1 Background to Stochastic Volatility 230

9.2 Heston Model in Brief 231

9.3 State Space Framework 232

9.3.1 Transition Equation 232

9.3.2 Measurement Equation: CDS Option Price 233

9.3.3 Measurement Equation Derivation 235

9.4 General State Space Model and Filter Revisited 237

9.4.1 Additive Non-Linear State Space model (Recap) 238

9.4.2 The Scaled Unscented Transformation (Recap) 240

9.5 The Application of Unscented Kalman Filter 243

9.6 Empirical Results 245

9.7 Concluding Remarks 249

10 Stochastic Volatility Model and Non-Linear Filtering Application

10.1 Background to Stochastic Volatility Models 258

10.2 Stochastic Volatility Models of Short-term Interest Rates 259

10.2.1 SV-ARMA Specification 261

10.2.2 Exogenous Variables 262

10.3 Data for Analysis 263

10.4 Analysis of Estimation Results 264

10.5 Comparison of Volatility Estimates 266

10.6 Outline of State Space Model Estimation via MCL 271

10.7 Concluding Summary 273

11 Applications for Filtering with Jumps

11.1 Background to Electricity Market and Prices 285

11.2 A Model for Spot Electricity Prices 288

11.3 State Space Model, Kalman Filter and Poisson Jumps 291

11.4 Data and Empirical Results for Electricity Market 294

11.5 Summarizing Electricity Market Application 296

11.6 Background to Jumps in CDS Indices 297

11.7 CDS Data and Preliminary Analysis 300

11.8 Methodology for Analyzing CDS Jump Risks 301

11.8.1 Normality Test for CDS Index Distribution 301

11.8.2 Model for Individual iTraxx Indices 301

11.8.3 Multivariate Analysis of Jumps in iTraxx Index with One Latent Common Factor 304

11.9 Analysis of Results from the CDS Market 307

11.10 Summarizing CDS Market Application 308

Bibliography 320

Index 33