The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives
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.
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
The Empirical problem
Estimating the volatility of the forward rates
Estimating the correlation structure
Estimating the volatility of the volatility
Hedging the Volatility Structure
Hedging the Correlation Structure
Hedging in conditions of market stress
I. THE THEORETICAL SET-UP.
2. The LIBOR Market Model.
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.
5.6 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.
6.9 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?
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?
11. Various Types of Hedging.
11.1 Statement of the Problem.
11.2 Three Types of Hedging.
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.
13.3 Hedging Results for the SABR Model.
13.4 Hedging Results for the LMM-SABR Model.
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.
15.6 Are We Getting Something for Nothing?
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.