Libor market models with stochastic volatility and CMS spread option pricing [Elektronische Ressource] / vorgelegt von Matthias Lutz

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2011
• Nombre de lectures: 154
• Langue: English
• Poids de l'ouvrage: 8 Mo

Extrait

Libor Market Models with Stochastic Volatility
and CMS Spread Option Pricing
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
vorgelegt von
Dipl.-Math. oec., M.Sc. Matthias Lutz
aus Wertheim
Ulm, im Februar 2011Amtierender Dekan: Prof. Dr. Paul Wentges
1. Gutachter: Prof. Dr. Rüdiger Kiesel
2. Prof. Dr. Ulrich Stadtmüller
3. Gutachter: Prof. Dr. John Schoenmakers
Tag der Promotion: 27.05.2011To the memory of Iris.Acknowledgments
First and foremost I would like to express my deepest gratitude to Prof. Dr. Rudiger
Kiesel for his con dence in my work and for giving me the freedom to follow my
interests. Moreover, I very much enjoyed being a member of the Institute of Math-
ematical Finance when he was chair of the institute. He created a pleasant working
environment and I always enjoyed teaching under his guidance.
I also would like to express my sincere thanks to Prof. Dr. Ulrich Stadtmuller
for being my co-examiner.
To my friends and colleagues, who are too numerous to mention individually:
Thank you for many fruitful discussions, your companionship and many great non-
university activities. Special thanks go to Christian Hering, Eva Nacca, Andreas
Rupp and Dennis Schat z for countless and \highly inspiring" co ee breaks. These
will be sorely missed.
I am also deeply indebted to Katrin Jensen and Andreas Rupp for proofreading
this thesis and for providing valuable comments.
Ulm, Februrary, 2011
Matthias Lutz
viiContents
1 Introduction 1
1.1 Fixed-Income Instruments . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective of the Thesis and Contribution . . . . . . . . . . . . . . . . 4
1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Interest-Rate Products and Pricing Models 7
2.1 Probabilistic Framework and Arbitrage-Free Pricing . . . . . . . . . . 7
2.2 The Yield Curve, Forwards & Swaps . . . . . . . . . . . . . . . . . . . 9
2.2.1 Zero-Coupon Bonds and Spot Rates . . . . . . . . . . . . . . . 9
2.2.2 Forward-Rate Agreements and Forward Rates . . . . . . . . . . 10
2.2.3 The Short Rate and Instantaneous Forward Rates . . . . . . . 11
2.2.4 Interest-Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.5 Yield-Curve Construction . . . . . . . . . . . . . . . . . . . . . 14
2.3 Fixed-Income Probability Measures . . . . . . . . . . . . . . . . . . . . 15
2.4 Caps, Floors & Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Vanilla Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.1 Black’s Model and the Volatility Smile . . . . . . . . . . . . . . 22
2.5.2 Local Volatility Models . . . . . . . . . . . . . . . . . . . . . . 25
2.5.3 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . 30
2.6 Other Interest-Rate Options . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 CMS Swaps, Caps and Floors . . . . . . . . . . . . . . . . . . . 39
2.6.2 CMS Spread Products . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.3 More Exotic Products . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Term-Structure Models: From Short-Rate Models to HJM . . . . . . . 53
2.7.1 Short-Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7.2 The HJM Framework . . . . . . . . . . . . . . . . . . . . . . . 57
3 The Libor Market Model 59
3.1 Model Set-Up and No-Arbitrage Dynamics . . . . . . . . . . . . . . . 60
3.1.1 Swap-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 A Stochastic-Volatility Extended LMM . . . . . . . . . . . . . . . . . 65
3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 Pricing European Options . . . . . . . . . . . . . . . . . . . . . 67
3.3 Parameterization of the LMM . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Volatility Structure . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.2 Skew Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
ixx CONTENTS
3.4.1 Pre-Calibration: E ective Swaption and Caplet Parameters . . 72
3.4.2 Main Calibration: Time-Dependent Parameters . . . . . . . . . 73
4 Efficient Pricing of CMS Spread Options 75
4.1 Approximating the Swap-Rate Dynamics . . . . . . . . . . . . . . . . 75
4.2 Iterated Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 The Density of the Integrated Variance . . . . . . . . . . . . . . . . . . 80
4.3.1 The Branch-Cut Corrected Laplace Transform . . . . . . . . . 80
4.3.2 Calculating the Bromwich Integral . . . . . . . . . . . . . . . . 81
4.3.3 The Optimal Linear Contour . . . . . . . . . . . . . . . . . . . 83
4.4 Calculating CMS Spread Option Prices . . . . . . . . . . . . . . . . . 86
4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.1 Time-Dependent Parameters . . . . . . . . . . . . . . . . . . . 90
4.5.2 Constant Parameters . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 New Correlation Parameterizations 97
5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.1.1 Historical Correlations . . . . . . . . . . . . . . . . . . . . . . . 98
5.1.2 Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Classical Parameterizations . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 New Flexible Correlation Parameterizations . . . . . . . . . . . . . . . 103
5.3.1 Alternative Characterization of the SC-Family . . . . . . . . . 104
5.3.2 Reformulation of the Cholesky Decomposition . . . . . . . . . . 105
5.3.3 New Parametric Forms . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Fitting Historical Correlations . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 DCT Rank-Reduced Parameterizations 115
6.1 Existing Methods for Rank-Reducing Correlation Matrices . . . . . . . 116
6.2 The DCT Rank Reduction Method . . . . . . . . . . . . . . . . . . . . 118
6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Extracting Correlations from the Market 125
7.1 Including CMS Spread Options . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Calibration Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.2 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3 Pricing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Appendices 141
A Laplace Transform of V (T ) . . . . . . . . . . . . . . . . . . . . . . . . 141
B Time-Dependent Parameter Scenario . . . . . . . . . . . . . . . . . . . 147
C Standard -extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 1481
D Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149