This book presents a major innovation in the interest rate space. It explains a financially motivated extension of the LIBOR Market model which accurately reproduces the prices for plain vanilla hedging instruments (swaptions and caplets) of all strikes and maturities produced by the SABR model. The authors show how to accurately recover the whole of the SABR smile surface using their extension of the LIBOR market model. This is not just a new model, this is a new way of option pricing that takes into account the need to calibrate as accurately as possible to the plain vanilla reference hedging instruments and the need to obtain prices and hedges in reasonable time whilst reproducing a realistic future evolution of the smile surface. It removes the hard choice between accuracy and time because the framework that the authors provide reproduces today’s market prices of plain vanilla options almost exactly and simultaneously gives a reasonable future evolution for the smile surface.
The authors take the SABR model as the starting point for their extension of the LMM because it is a good model for European options. The problem, however with SABR is that it treats each European option in isolation and the processes for the various underlyings (forward and swap rates) do not talk to each other so it isn’t obvious how to relate these processes into the dynamics of the whole yield curve. With this new model, the authors bring the dynamics of the various forward rates and stochastic volatilities under a single umbrella. To ensure the absence of arbitrage they derive drift adjustments to be applied to both the forward rates and their volatilities. When this is completed, complex derivatives that depend on the joint realisation of all relevant forward rates can now be priced.
Contents
THE THEORETICAL SET-UP
The Libor Market model
The SABR Model
The LMM-SABR Model
IMPLEMENTATION AND CALIBRATION
Calibrating the LMM-SABR model to Market Caplet prices
Calibrating the LMM/SABR model to Market Swaption Prices
Calibrating the Correlation Structure
EMPIRICAL EVIDENCE
The Empirical problem
Estimating the volatility of the forward rates
Estimating the correlation structure
Estimating the volatility of the volatility
HEDGING
Hedging the Volatility Structure
Hedging the Correlation Structure
Hedging in conditions of market stress
विषयसूची
Acknowledgements xi
1 Introduction 1
I The Theoretical Set-Up 7
2 The LIBOR Market Model 9
3 The SABR Model 25
4 The LMM-SABR Model 51
II Implementation and Calibration 79
5 Calibrating the LMM-SABR Model to Market Caplet Prices 81
6 Calibrating the LMM-SABR Model to Market Swaption Prices 101
7 Calibrating the Correlation Structure 125
III Empirical Evidence 141
8 The Empirical Problem 143
9 Estimating the Volatility of the Forward Rates 159
10 Estimating the Correlation Structure 181
IV Hedging 203
11 Various Types of Hedging 205
12 Hedging against Moves in the Forward Rate and in the Volatility 221
13 (LMM)-SABR Hedging in Practice: Evidence from Market Data 231
14 Hedging the Correlation Structure 247
15 Hedging in Conditions of Market Stress 257
References 271
Index 275
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RICCARDO REBONATO is Global Head of Market Risk and Global Head of the Quantitative Research Team at RBS. He is a visiting lecturer at Oxford University (Mathematical Finance) and adjunct professor at Imperial College (Tanaka Business School). He sits on the Board of Directors of ISDA and on the Board of Trustees for GARP. He is an editor for the International Journal of Theoretical and Applied Finance, for Applied Mathematical Finance, for the Journal of Risk and for the Journal of Risk Management in Financial Institutions. He holds doctorates in Nuclear Engineering and in Science of Materials/Solid State Physics. He was a research fellow in Physics at Corpus Christi College, Oxford, UK.
KENNETH MCKAY is a Ph D student at the London School of Economics following a first class honours degree in Mathematics and Economics from the LSE and an MPhil in Finance from Cambridge University. He has been working on interest rate derivative-related research with Riccardo Rebonato for the past year.
RICHARD WHITE holds a doctorate in Particle Physics from Imperial College London, and a first class honours degree in Physics from Oxford University. He held a Research Associate position at Imperial College before joining RBS in 2004 as a Quantitative Analyst. His research interests include option pricing with Levy Processes, Genetic Algorithms for portfolio optimisation, and Libor Market Models with stochastic volatility. He is currently taking a fortuitously timed sabbatical to pursue his joint passion for travel and scuba diving.
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