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1．General Introduction,Changing Volatility Models and European Options Pricing1.1 General Introduction 1

1.2 Introduction to Changing Volatility Models 4

1.3 Model Completeness and European Option Pricing 4

1.4 Single Period Volatility Changing Problems 9

1.4.1 Fixed Volatility Changing Time with Barrier B 9

1.4.2 Random Volatility Changing Time with a Hitting Barrier B 14

1.5 Multi-Period Volatility Changing Problems 15

1.6 Extension to Incomplete Market 19

1.6.1 A Simple Random Volatility Changing Model-Extension to Stochastic Volatility Model 19

1.6.2 Future Research 20

1.7 Appendix:Proof 20

1.7.1 Proof of Proposition 1.1 20

1.7.2 Proof of Proposition 1.2 23

1.7.3 Proof of Proposition 1.3 24

1.7.4 Proof of Proposition 1.4 26

1.7.5 Proof of Proposition 1.5 27

1.7.6 Proof of Proposition 1.6 29

1.7.7 Theorem 2.2 of[132]:Uniqueness of the Equivalent Martingale Measure 30

2．Introduction to Stochastic Volatility and Local Stochastic Volatility Models2.1 Stochastic Volatility Models-A General Set-Up 31

2.1.1 Model Set-Up 31

2.1.2 Change of Measure and Model Incompleteness 32

2.2 Making the Stochastic Volatility Economy Complete 34

2.3 European Option Price 36

2.4 Local Stochastic Volatility Models:An Introduction 38

2.5 Adjustment to the Calculation of Greeks in a Non-Constant Implied Volatility Model 42

3．Foreign Exchange Options with Local Stochastic Volatility and Stochastic Interest Rates3.1 Introduction 44

3.2 The FX-IR Hybrid Model 46

3.3 Asymptotic Expansion 50

3.3.1 A Brief Introduction 50

3.3.2 European Option Pricing and Implied Volatility 50

3.4 Model Implementation and Numerical Results 58

3.5 FX Option Pricing via Fourier Transform under Stochastic Interest Rates,Stochastic Volatility and the Jump Process 66

3.5.1 The Multi-Factor Model 66

3.5.2 Change of Measure and Option Pricing 67

3.5.3 Model Implementation 70

3.5.4 Calibration Results for the USD/JPY Market 72

3.6 Perfect Hedging with Stochastic Interest Rates and Local Stochastic Volatility 75

3.6.1 Hedging with Options 76

3.6.2 Hedging with Options and Bonds 80

3.7 Partial Hedging with Hedging Error Analysis 82

3.7.1 Hedging with One Option for the Volatility Risk 83

3.7.2 Hedging with One Option for the Interest Rate Risk 85

3.8 Model Mis-specification and Hedging Error Analysis 86

3.8.1 Delta Hedging Difference between the CEV and CEV-SV Models 86

3.8.2 Model Mis-specification:The Importance of Stochastic Interest Rates 88

3.9 Application to Power-Reverse-Dual-Currency Notes 91

3.9.1 PRDC-TARN:The Structured Product 91

3.9.2 Smile Impact on PRDC-TARN Product Valuation 93

3.10 Conclusion and Future Research 95

3.11 Appendix:Proof 96

3.11.1 Derivation of the European Option Formula 96

3.11.2 Conditional Expectations of the Multiple Weiner-Ito Integral 101

3.11.3 Watanabe Theorem 104

3.11.4 The European Option Formula from the Fourier Transform Method 105

4．Non-Biased Monte Carlo Simulation for a Heston-Type Stochastic Volatility Model4.1 Introduction 107

4.2 Properties of the Square Root Process 108

4.3 Simulation of Vt:Application of the Saddle Point Method 111

4.4 Simulation of ∫t+△ t Vs ds given Vt and Vt+△:Moment Matching Technique 112

4.5 Simulation of Ft+△t given Ft 113

4.6 Conclusion and Future Research 114

4.7 Appendix:Proof 114

4.7.1 Characteristic Function and Moments of the Square Root Process 114

4.7.2 The Lugannani and Rice Formula for the Cumulative Distribution Function 116

5．The LIBOR Market Model with Stochastic Volatility and Jump Processes5.1 Introduction 118

5.2 The LIBOR Forward Rate Model 121

5.2.1 Risk-Neutral Measure 121

5.2.2 Change of Measure 123

5.3 The LIBOR Swap Rate Model 124

5.3.1 The Swap Market 124

5.3.2 Change of Measure 125

5.4 Caplet and Swaption Pricing Via Fourier Transform 128

5.4.1 Caplet Pricing 128

5.4.2 Swaption Pricing 130

5.5 Conclusion and Future Research 132

5.6 Appendix:Proof 133

5.6.1 Proof of Lemma 5.1 133

5.6.2 Proof of Proposition 5.2 134

5.6.3 Proof of Proposition 5.3 136

5.6.4 Proof of Proposition 5.4 137

5.6.5 The Marked Point Process 138

5.6.6 The Girsanov Theorem and Ito's Lemma on Jump Processes The Girsanov Theorem for Jump Processes 139

5.6.7 The Derivation of Characteristic Functions Caplet 142

Bibliography 146