Richard White & Riccardo Rebonato 
The SABR/LIBOR Market Model 
Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives

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

Table of Content

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

About the author

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.