Interest-rate Models: an Extension to the Usage in the Energy Market and Pricing Exotic Energy Derivatives Dan Wang Supervisor: Professor Nigel Meade Center for Quantitative Finance The Business School Imperial College of Science, Technology and Medicine In partial fulfillment of the requirements for the degree of Doctor of Philosophy May, 2009 Abstract In this thesis, we review various popular pricing models in the interest-rate market. Among these pricing models, we choose the LIBOR Market model (LMM) as the benchmark model. Based on market practice experience, we also develop a pricing model named the “Market volatility model”. By pricing vanilla interest-rate options such as interest-rate caps and swaptions, we compare the performance of our Market volatility model to that of the LMM. It is proved that the Market Volatility model produce comparable results to the LMM, while its computing efficiency largely exceeds that of the LMM. Following the recent rapid development in the commodity market, in particular the energy market, we attempt to extend the use of our proposed Market volatility model from the interest-rate market to the energy market. We prove that the Market Volatility model is capable of pricing various energy derivative under the assumption of absence of the convenience yield. In addition, we propose a new type of exotic energy derivative which has a flexible option structure. This energy derivative is named as the Flex-Asian spread options (FASO). We give examples of different option structures within the FASO framework and use the Market volatility model to generate option prices and greeks for each structure. Although the Market volatility model can be used to price various energy derivatives based on oil/gas contracts, it is not compatible with the structure of one of the most advanced derivatives in the energy market, the storage option. We modify the existing pricing model for storage options and use our own 3D-binomial tree approach to price gas storage contracts. By doing these, we improve the performance of the traditional storage model. ii Acknowledgment I would like to thank my supervisor, Professor Nigel Meade and Professor Nicos Christofides, for their guidance through my PhD research and the writing of this thesis. I would also like to thank my current manager Philip Atkins at BP and my previous manager Maarten Wiersinga at Mizuho International, for offering me the invaluable opportunity to work in their teams. Thanks to all my previous and current colleagues from the Risk Quantitative Analysis team at BP and the Model Validation team at Mizuho respectively, for helping me to learn practical knowledge in the financial markets. I would like to thank all my previous teacher, friends and colleagues, for those of you who have helped me at work and in life. In particular, I would like to thank Yukun Shen, for all your invaluable inspirations and help when I worked on this thesis. I would also like to thank Lin Luo, for bringing up the idea of pursuing a PhD at CQF; Maggie Zhang and Carole Ye, for providing great help when I was working at Mizuho; Jing Wei, for the on-going support and being there whenever I was in need. I would like to thank my parents, who encourage me to face the challenges along the years of my PhD. Without your guidance and support, I could achieve nothing today. After all those years, I guess I have finally made some effort and I just wish I could make you proud. This thesis is dedicated to my parents with my love. iii Contents 1. Introduction 8 2. Interest-Rate Models 11 2.1. The LIBOR Market Model................................ 11 2.1.1. The Heath-Jarrow-Morton (HJM) framework.................. 12 2.1.2. Log-normal LIBOR Models............................ 14 2.1.2.1. Determining the No-Arbitrage Drifts of Forward Rates....... 14 2.1.2.2. The Miltersen-Sandmann-Sondermann Approach........... 18 2.1.2.3. The Brace-Gatarek-Musiela Approach................. 18 2.1.2.4. The Musiela-Rutkowski Approach................... 20 2.1.3. Descriptions of the Forward Rate Dynamics of the LIBOR Market Model.. 24 2.1.3.1. The First Description of Forward LIBOR Rates........... 24 2.1.3.2. The Second Description of Forward LIBOR Rates.......... 25 2.1.3.3. The Third Description of Forward LIBOR Rates........... 26 2.1.3.4. Principal Component Analysis (PCA)................. 27 2.1.4. Finding the Suitable Functional Forms for Instantaneous Volatilities..... 28 2.1.5. Calibration of the Caplet and Swaption Volatilities in LMM.......... 31 2.1.5.1. Calibration for Caplet Volatilities................... 32 2.1.5.2. Calibration for European Swaption Volatilities............ 33 2.1.6. Calibration for Instantaneous Correlations.................... 34 2.1.7. Incompatibility between the LSM and the LFM................ 37 2.2. The Stochastic-αβρ (SABR) model............................ 38 2.2.1. The Stochastic-β Model Framework....................... 38 2.2.1.1. Implied Normal Volatility....................... 39 2.2.1.2. Implied Black Volatility......................... 39 2.2.2. Pricing Formula for a Vanilla Call Option.................... 40 2.3. The CIR-Jump model................................... 40 2.4. The Market Volatility Model............................... 42 2.4.1. Interpolation of the Piecewise-Constant Volatilities of Forward Rates..... 43 2.4.1.1. The “Stripping” Method for the Caplet Volatilities......... 43 2.4.1.2. The Bi-linear Interpolation for the Swaption Volatilities....... 45 2.4.2. The link between the Market Volatility Model and the Implied Trinomial Tree Method.................................... 47 2.5. Summary.......................................... 48 iv Contents 3. Pricing Interest-rate Options 50 3.1. European-style Interest-Rate Options........................... 50 3.1.1. Interest-Rate Caps/Floors............................. 50 3.1.2. European Swaptions................................ 52 3.2. Exotic Interest-Rate Options............................... 53 3.2.1. Constant Maturity Swaps............................. 53 3.2.2. CMS Spread Options................................ 55 3.3. Performance of LMM v.s. the Market Volatility Model................. 56 3.3.1. European Swaptions................................ 56 3.3.2. Caps......................................... 63 3.4. Pricing Exotic Interest-Rate Options Using the Market Volatility Model....... 67 3.5. Summary.......................................... 71 4. From the Interest-Rate Market to the Energy Market 73 4.1. Introduction of the Energy Market............................ 73 4.1.1. The Oil Market................................... 73 4.1.2. The Natural Gas Market............................. 74 4.2. The Spot-Forward Relationship in the Energy/Commodity Market.......... 76 4.3. Common Pricing Models in the Energy Market..................... 79 4.3.1. The Bachelier Model............................... 79 4.3.2. The Black-Scholes Model............................. 79 4.3.3. The Mean-Reverting Model............................ 80 4.3.4. The Jump-Diffusion Model............................ 80 4.3.5. The Stochastic Volatility Model.......................... 81 4.4. The Market Volatility Model for the Energy Market.................. 81 4.5. Summary.......................................... 82 5. Pricing Exotic Energy Options 83 5.1. Spread Options....................................... 83 5.2. Basket Options....................................... 84 5.3. Asian Options........................................ 84 5.4. Flex-Asian Spread Options (FASO)............................ 85 5.4.1. The FASO Type-1 Option............................. 86 5.4.2. The FASO Type-2 Option............................. 86 5.4.3. The FASO Type-3 Option............................. 87 5.4.4. The FASO Type-4 Option............................. 88 5.5. Test Scenarios for FASO.................................. 89 5.5.1. Moneyness Definition............................... 89 5.5.2. Structures of the Pricing Windows in FASO................... 91 5.5.3. Correlation Matrices................................ 95 5.5.4. Volatilities..................................... 96 5.5.5. Test Results..................................... 96 5.6. Summary.......................................... 101 v Contents 6. The Storage Model 103 6.1. The Intuition Behind the Development of Storage Option Pricing Models...... 103 6.2. Various Methods for Valuing a Storage Contract.................... 104 6.2.1. The Rolling Intrinsic Method........................... 106 6.2.2. The Trinomial Tree Forest Method........................ 107 6.2.3. The Basket of Spread Options Method...................... 107 6.2.3.1. The Rolling Basket of Spread Options Method............ 108 6.2.4. The Least Squares Monte-Carlo Method..................... 109 6.3. The Storage Model Methodology............................. 111 6.3.1. Brief Introduction of the Calibration Process.................. 112 6.3.2. Modeling volatilities................................ 113 6.3.3. Modeling correlation................................ 118 6.3.3.1. Local Correlation............................ 118 6.3.3.2. Term Correlation............................ 119 6.3.3.3. Cash Correlation............................. 119 6.3.3.4. Correlation Analysis........................... 120 6.3.4. Calendar Spread Option Pricing Model..................... 123 6.3.5. The Analytic Formulas of Spread Option Greeks for the Storage Model... 124 6.3.6. Linear Programming Problem........................... 126 6.4. Storage values.......................................