免費 A Stochastic Volatility LIBOR Market Model with a closed form solution

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A Stochastic Volatility LIBOR Market Model with a closed form solution
by Hazim Nada

Imperial College London
Centre for Quantitative Finance

1 Introduction 6
2 Literature Review 9
2.1 Review of Term Structure Models . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 The HJM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 The Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Implied Volatility and Market Models . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 The Volatility Smile Problem . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Local Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Stochastic Volatility LIBOR Market Models . . . . . . . . . . . . . 37
2.3 Summary and overture to the next chapter . . . . . . . . . . . . . . . . . . 44
3 An alternative Stochastic Volatility LMM 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Motivations and Model Layout . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Model’s Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Properties of the Model’s Implied Volatility . . . . . . . . . . . . . . . . . 59
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Pricing of Caps and Swaptions 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.5 Summary and Overture to the next chapter . . . . . . . . . . . . . . . . . 87
5 Calibration and Empirical Fit of Prices 89
5
5.1 Introduction to methodology and data . . . . . . . . . . . . . . . . . . . . 89
5.2 Caplets Calibration and comparison with SABR . . . . . . . . . . . . . . . 96
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Comparative Analysis with Black’s Model 113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Prices Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Conclusion 136
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2 Suggestions For Further Research . . . . . . . . . . . . . . . . . . . . . . . 139
8 Appendices 140
8.1 Appendix to Chapter 2 (Basics of Stochastics and Pricing Theory) . . . . . 140
8.2 Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.3 Appendix to Chapter 4 (Implementation of Prices and Greeks) . . . . . . . 152
9 References 164

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关键词：Volatility Stochastic Solution Stochast solutio model Solution market form Volatility

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