This book presents a major innovation in the interest rate space.It explains a financially motivated extension of the LIBOR Marketmodel which accurately reproduces the prices for plain vanillahedging instruments (swaptions and caplets) of all strikes andmaturities produced by the SABR model. The authors show how toaccurately recover the whole of the SABR smile surface using theirextension of the LIBOR market model. This is not just a new model,this is a new way of option pricing that takes into account theneed to calibrate as accurately as possible to the plain vanillareference hedging instruments and the need to obtain prices andhedges in reasonable time whilst reproducing a realistic futureevolution of the smile surface. It removes the hard choice betweenaccuracy and time because the framework that the authors providereproduces today's market prices of plain vanilla options almostexactly and simultaneously gives a reasonable future evolution forthe smile surface.The authors take the SABR model as the starting point for theirextension of the LMM because it is a good model for Europeanoptions. The problem, however with SABR is that it treats eachEuropean option in isolation and the processes for the variousunderlyings (forward and swap rates) do not talk to each other soit isn't obvious how to relate these processes into the dynamics ofthe whole yield curve. With this new model, the authors bring thedynamics of the various forward rates and stochastic volatilitiesunder a single umbrella. To ensure the absence of arbitrage theyderive drift adjustments to be applied to both the forward ratesand their volatilities. When this is completed, complex derivativesthat depend on the joint realisation of all relevant forward ratescan now be priced.ContentsTHE THEORETICAL SET-UPThe Libor Market modelThe SABR ModelThe LMM-SABR ModelIMPLEMENTATION AND CALIBRATIONCalibrating the LMM-SABR model to Market Caplet pricesCalibrating the LMM/SABR model to Market Swaption PricesCalibrating the Correlation StructureEMPIRICAL EVIDENCEThe Empirical problemEstimating the volatility of the forward ratesEstimating the correlation structureEstimating the volatility of the volatilityHEDGINGHedging the Volatility StructureHedging the Correlation StructureHedging in conditions of market stress
|Product dimensions:||6.90(w) x 9.70(h) x 1.00(d)|
About the Author
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
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 ofVolatility, _.
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 CapletPrices.
5.1 The Caplet-Calibration Problem.
5.2 Choosing the Parameters of the Function, g (_), and theInitial.
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 SABRModel.
5.7 Implications for Model Choice.
6. Calibrating the LMM-SABR model to Market SwaptionPrices.
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 ofVolatility of the Swap Rate, V.
6.6 Approximating the Swap-Rate/Swap-Rate-VolatilityCorrelation, RSABR.
6.7 Approximating the Swap Rate Exponent, B.
6.9 Conclusions and Suggestions for Future Work.
6.10 Appendix: Derivation of Approximate Swap RateVolatility.
6.11 Appendix: Derivation of Swap-Rate/Swap-Rate-VolatilityCorrelation, 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 OptionsMarket.
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 theUnderlyings.
11.5 Second-Order Derivatives with Respect to theUnderlyings.
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 theVolatility.
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 MarketData.
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?
Most Helpful Customer Reviews
Having to deal with Exotic Interest Rates product professionally, I had to get the latest Rebonato's opum. I've found in the past that there is much to be annoyed with this author, but also very frequently insights you would not get anywhere else: in the case of this book, the couple of pages where he explains what makes a good model should be mandatory reading for any aspiring "quant" thinking about applying the tools of his trade to the dirty world of finance. Recommended as such.