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

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2009-04-13
  • Publisher: Wiley
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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

The Theoretical Set-Up.
The LIBOR Market Model.
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
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
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
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?
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?
Various Types of Hedging.
Statement of the Problem
Three Types of Hedging
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
Hedging Results for the SABR Model
Hedging Results for the LMM-SABR Model
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
Are We Getting Something for Nothing?
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