The SABR/LIBOR Market Model Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives

by Rebonato, Riccardo; McKay, Kenneth; White, Richard

• ISBN13: 9780470740057
• ISBN10: 0470740051
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-04-13
• Publisher: Wiley

Summary

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

Author Biography

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 PhD 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.

Table of Contents

Introduction.

The Theoretical Set-Up.

The LIBOR Market Model.

Definitions

The Volatility Functions

Separating the Correlation from the Volatility Term

The Caplet-Pricing Condition Again

The Forward-Rate/Forward-Rate Correlation

Possible Shapes of the Doust Correlation Function

The Covariance Integral Again

The SABR Model.

The SABR Model (and Why It Is a Good Model

Description of the Model

The Option Prices Given by the SABR Model

Special Cases

Qualitative Behaviour of the SABR Model

The Link Between the Exponent, _, and the Volatility of Volatility, _

Volatility Clustering in the (LMM)-SABR Model

The Market

How Do We Know that the Market Has Chosen _ = 0:5?

The Problems with the SABR Model

The LMM-SABR Model.

The Equations of Motion

The Nature of the Stochasticity Introduced by Our Model

A Simple Correlation Structure

A More General Correlation Structure

Observations on the Correlation Structure

The Volatility Structure

What We Mean by Time Homogeneity

The Volatility Structure in Periods of Market Stress

A More General Stochastic Volatility Dynamics

Calculating the No-Arbitrage Drifts

IMPLEMENTATION AND CALIBRATION.

Calibrating the LMM-SABR model to Market Caplet Prices.

The Caplet-Calibration Problem

Choosing the Parameters of the Function, g (_), and the Initial Values, kT 0

Choosing the Parameters of the Function h(_

Choosing the Exponent, _, and the Correlation, _SABR

Results

Calibration in Practice: Implications for the SABR Model

Implications for Model Choice

Calibrating the LMM-SABR model to Market Swaption Prices.

The Swaption Calibration Problem

Swap Rate and Forward Rate Dynamics

Approximating the Instantaneous Swap Rate Volatility, St

Approximating the Initial Value of the Swap Rate Volatility, _0 (First Route

Approximating _0

Approximating the Swap-Rate/Swap-Rate-Volatility Correlation, RSABR

Approximating the Swap Rate Exponent, B

Results

Conclusions and Suggestions for Future Work

Appendix: Derivation of Approximate Swap Rate Volatility

Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation, RSABR

Appendix: Approximation of

Calibrating the Correlation Structure.

Statement of the Problem

Creating a Valid Model Matrix

A Case Study: Calibration Using the Hypersphere Method

Which Method Should One Choose?

Appendix1

EMPIRICAL EVIDENCE.

The Empirical Problem.

Statement of the Empirical Problem

What Do We know from the Literature?

Data Description

Distributional Analysis and Its Limitations

What Is the True Exponent _?

Appendix: Some Analytic Results

Estimating the Volatility of the Forward Rates.

Expiry-Dependence of Volatility of Forward Rates

Direct Estimation

Looking at the Normality of the Residuals

Maximum-Likelihood and Variations on the Theme

Information About the Volatility from the Options Market

Overall Conclusions

Estimating the Correlation Structure.

What We Are Trying To Do

Some Results from Random Matrix Theory

Empirical Estimation

Descriptive Statistics

Signal and Noise in the Empirical Correlation Blocks

What Does Random Matrix Theory Really Tell Us?

Calibrating the Correlation Matrices

How Much Information Do the Proposed Models Retain?

HEDGING.

Various Types of Hedging.

Statement of the Problem

Three Types of Hedging

Definitions

First-Order Derivatives with Respect to the Underlyings

Second-Order Derivatives with Respect to the Underlyings

Generalizing Functional-Dependence Hedging

How Does the Model Know about Volga and Vanna?

Choice of Hedging Instrument

Hedging Against Moves in the Forward Rate and in the Volatility.

Delta Hedging in the SABR-(LMM) Model

Vega Hedging in the SABR-(LMM) Model

(LMM)-SABR Hedging in Practice: Evidence from Market Data.

Purpose of this Chapter

Notation

Hedging Results for the SABR Model

Hedging Results for the LMM-SABR Model

Conclusions

Hedging the Correlation Structure.

The Intuition Behind the Problem

Hedging the Forward-Rate Block

Hedging the Volatility-Rate Block

Hedging the Forward-Rate/Volatility Block

Final Considerations

Hedging in Conditions of Market Stress.

Statement of the Problem

The Volatility Function

The Case Study

Hedging

Results

Are We Getting Something for Nothing?

Table of Contents provided by Publisher. All Rights Reserved.