Riccardo Rebonato, Kenneth McKay, Richard White

The SABR/LIBOR Market Model

• Gebundenes Buch

Produktbeschreibung

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.

Produktdetails

• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 304
• Erscheinungstermin: 1. April 2009
• Englisch
• Abmessung: 250mm x 175mm x 21mm
• Gewicht: 665g
• ISBN-13: 9780470740057
• ISBN-10: 0470740051
• Artikelnr.: 25656778

Autorenporträt

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.

Inhaltsangabe

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?