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


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.

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.

1. Introduction.


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.


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.


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?


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?