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Description of The SABR/LIBOR Market Model

The authors take two market standards, the SABR and the LIBOR Market Model (LMM) and produce a coherent synthesis for the pricing of complex interest rate derivatives. The SABR model has become the market standard to recover the price of European options. Its main strengths are its financial justifiability, and its ability to recover the dynamics of the smile evolution when the underlying changes. However, the SABR model treats each European option in isolation. The processes for forward rates and swap rates cannot easily be combined to create coherent dynamics for the entire yield curve. With their new model, the authors bring the dynamics of the various forward rates and stochastic volatilities under a single measure, and derive 'drift adjustments' to ensure the absence of arbitrage and to allow for the pricing of complex derivatives. The credible evolution of future smiles generated by the model is essential to complex derivatives pricing as it determines future prices for caplets and swaptions and therefore plausible re–hedging costs. The authors calibrate their model to hedging instruments in a way that is both accurate and extremely simple. They also propose a pragmatic hedging approach, inspired by work done with the two–state Markov–chain approach which relies on the empirical regularities of the dynamics of the smile surface and the robustness of the fits proposed. The final chapter considers ‘survival’ hedging in times of market turmoil. It does so by providing a set of transactions that can protect the value of a complex derivatives book in a stressed market.

The extension of the LMM model provides a valid description of the financial reality while retaining tractability, computational speed and ease of calibration. The goal for the new model is to offer the ability to reduce uncertainty in market prices to an acceptable minimum by making as judicious a use as possible of the econometric information available. The grounding in empirical information of the modelling approach utilised by the authors differentiates this title from the stochastic–calculus–heavy, but empirically light, work of others.

The title will be of interest to quantitative analysts, quantitative developers, risk managers and traders in complex derivatives.

Title Information

ISBN:

9780470740057

Pages:

296 pages

Format:

Hardback

Product Code:

321372

Publisher:

John Wiley & Sons Ltd

Published:

06/03/2009

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About Riccardo Rebonato and Kenneth McKay and Richard White

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.

Contents of The SABR/LIBOR Market Model

1. Introduction

I. THE THEORETICAL SET-UP

2. The LIBOR Market Model
2.1 Definitions
2.2 The Volatility Functions
2.3 Separating the Correlation from the Volatility Term
2.4 The Caplet-Pricing Condition Again
2.5 The Forward-Rate/Forward-Rate Correlation
2.6 Possible Shapes of the Doust Correlation Function
2.7 The Covariance Integral Again

3. The SABR Model
3.1 The SABR Model (and Why It Is a Good Model
3.2 Description of the Model
3.3 The Option Prices Given by the SABR Model
3.4 Special Cases
3.5 Qualitative Behaviour of the SABR Model
3.6 The Link Between the Exponent, _, and the Volatility of Volatility, _
3.7 Volatility Clustering in the (LMM)-SABR Model
3.8 The Market
3.9 How Do We Know that the Market Has Chosen _ = 0:5?
3.10 The Problems with the SABR Model

4. The LMM-SABR Model
4.1 The Equations of Motion
4.2 The Nature of the Stochasticity Introduced by Our Model
4.3 A Simple Correlation Structure
4.4 A More General Correlation Structure
4.5 Observations on the Correlation Structure
4.6 The Volatility Structure
4.7 What We Mean by Time Homogeneity
4.8 The Volatility Structure in Periods of Market Stress
4.9 A More General Stochastic Volatility Dynamics
4.10 Calculating the No-Arbitrage Drifts


II. IMPLEMENTATION AND CALIBRATION

5 Calibrating the LMM-SABR model to Market Caplet Prices
5.1 The Caplet-Calibration Problem
5.2 Choosing the Parameters of the Function, g (_), and the Initial
Values, kT 0
5.3 Choosing the Parameters of the Function h(_
5.4 Choosing the Exponent, _, and the Correlation, _SABR
5.5 Results
5.6 Calibration in Practice: Implications for the SABR Model
5.7 Implications for Model Choice

6. Calibrating the LMM-SABR model to Market Swaption Prices
6.1 The Swaption Calibration Problem
6.2 Swap Rate and Forward Rate Dynamics
6.3 Approximating the Instantaneous Swap Rate Volatility, St
6.4 Approximating the Initial Value of the Swap Rate Volatility, _0 (First Route
6.5 Approximating _0 (Second Route and the Volatility of Volatility of the Swap Rate, V
6.6 Approximating the Swap-Rate/Swap-Rate-Volatility Correlation, RSABR
6.7 Approximating the Swap Rate Exponent, B
6.8 Results
6.9 Conclusions and Suggestions for Future Work
6.10 Appendix: Derivation of Approximate Swap Rate Volatility
6.11 Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation, RSABR
6.12 Appendix: Approximation of

7. Calibrating the Correlation Structure
7.1 Statement of the Problem
7.2 Creating a Valid Model Matrix
7.3 A Case Study: Calibration Using the Hypersphere Method
7.4 Which Method Should One Choose?
7.5 Appendix1


III. EMPIRICAL EVIDENCE

8. The Empirical Problem
8.1 Statement of the Empirical Problem
8.2 What Do We know from the Literature?
8.3 Data Description
8.4 Distributional Analysis and Its Limitations
8.5 What Is the True Exponent _?
8.6 Appendix: Some Analytic Results

9. Estimating the Volatility of the Forward Rates
9.1 Expiry-Dependence of Volatility of Forward Rates
9.2 Direct Estimation
9.3 Looking at the Normality of the Residuals
9.4 Maximum-Likelihood and Variations on the Theme
9.5 Information About the Volatility from the Options Market
9.6 Overall Conclusions

10. Estimating the Correlation Structure
10.1 What We Are Trying To Do
10.2 Some Results from Random Matrix Theory
10.3 Empirical Estimation
10.4 Descriptive Statistics
10.5 Signal and Noise in the Empirical Correlation Blocks
10.6 What Does Random Matrix Theory Really Tell Us?
10.7 Calibrating the Correlation Matrices
10.8 How Much Information Do the Proposed Models Retain?


IV. HEDGING

11. Various Types of Hedging
11.1 Statement of the Problem
11.2 Three Types of Hedging
11.3 Definitions
11.4 First-Order Derivatives with Respect to the Underlyings
11.5 Second-Order Derivatives with Respect to the Underlyings
11.6 Generalizing Functional-Dependence Hedging
11.7 How Does the Model Know about Volga and Vanna?
11.8 Choice of Hedging Instrument

12. Hedging Against Moves in the Forward Rate and in the Volatility
12.1 Delta Hedging in the SABR-(LMM) Model
12.2 Vega Hedging in the SABR-(LMM) Model

13. (LMM)-SABR Hedging in Practice: Evidence from Market Data
13.1 Purpose of this Chapter
13.2 Notation
13.3 Hedging Results for the SABR Model
13.4 Hedging Results for the LMM-SABR Model
13.5 Conclusions

14. Hedging the Correlation Structure
14.1 The Intuition Behind the Problem
14.2 Hedging the Forward-Rate Block
14.3 Hedging the Volatility-Rate Block
14.4 Hedging the Forward-Rate/Volatility Block
14.5 Final Considerations

15. Hedging in Conditions of Market Stress
15.1 Statement of the Problem
15.2 The Volatility Function
15.3 The Case Study
15.4 Hedging
15.5 Results
15.6 Are We Getting Something for Nothing?