This ebook presents a serious innovation within the rate of interest house. It explains a financially motivated extension of the LIBOR Market mannequin which precisely reproduces the costs for plain vanilla hedging devices (swaptions and caplets) of all strikes and maturities produced by the SABR mannequin. The authors present the best way to precisely get better the entire of the SABR smile floor utilizing their extension of the LIBOR market mannequin. This isn’t just a brand new mannequin, this can be a new means of choice pricing that takes under consideration the necessity to calibrate as precisely as attainable to the plain vanilla reference hedging devices and the necessity to receive costs and hedges in cheap time while reproducing a practical future evolution of the smile floor. It removes the arduous alternative between accuracy and time as a result of the framework that the authors present reproduces immediately’s market costs of plain vanilla choices nearly precisely and concurrently provides an affordable future evolution for the smile floor.
The authors take the SABR mannequin as the start line for his or her extension of the LMM as a result of it’s a good mannequin for European choices. The drawback, nevertheless with SABR is that it treats every European choice in isolation and the processes for the assorted underlyings (ahead and swap charges) don’t speak to one another so it isn’t apparent the best way to relate these processes into the dynamics of the entire yield curve. With this new mannequin, the authors deliver the dynamics of the assorted ahead charges and stochastic volatilities underneath a single umbrella. To make sure the absence of arbitrage they derive drift changes to be utilized to each the ahead charges and their volatilities. When that is accomplished, advanced derivatives that rely on the joint realisation of all related ahead charges can now be priced.
THE THEORETICAL SET-UP
The Libor Market mannequin
The SABR Model
The LMM-SABR Model
IMPLEMENTATION AND CALIBRATION
Calibrating the LMM-SABR mannequin to Market Caplet costs
Calibrating the LMM/SABR mannequin to Market Swaption Prices
Calibrating the Correlation Structure
The Empirical drawback
Estimating the volatility of the ahead charges
Estimating the correlation construction
Estimating the volatility of the volatility
Hedging the Volatility Structure
Hedging the Correlation Structure
Hedging in circumstances of market stress
Table of Contents
- THE THEORETICAL SET-UP.
- 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.
- 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 within 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.
- The LMM-SABR Model.
4.1 The Equations of Motion.
4.2 The Nature of the Stochasticity Introduced by Our Model.
4.Three A Simple Correlation Structure.
4.Four 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.
- IMPLEMENTATION AND CALIBRATION.
5 Calibrating the LMM-SABR mannequin 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.
- Calibrating the LMM-SABR mannequin 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.
- Calibrating the Correlation Structure.
7.1 Statement of the Problem.
7.2 Creating a Valid Model Matrix.
7.Three A Case Study: Calibration Using the Hypersphere Method.
7.4 Which Method Should One Choose?
III. EMPIRICAL EVIDENCE.
- 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.
- Estimating the Volatility of the Forward Rates.
9.1 Expiry-Dependence of Volatility of Forward Rates.
9.2 Direct Estimation.
9.3 Looking on 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.
- 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 within 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?
- 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.
- Hedging Against Moves within the Forward Rate and within the Volatility.
12.1 Delta Hedging within the SABR-(LMM) Model.
12.2 Vega Hedging within the SABR-(LMM) Model.
- (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.
- 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.
- 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?
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 analysis fellow in Physics at Corpus Christi College, Oxford, UK.
Kenneth McKay is a PhD scholar on the London School of Economics following a firstclass honours diploma in Mathematics and Economics from the LSE and an MPhil in Finance from Cambridge University. He has been engaged on rate of interest derivative-related analysis with Riccardo Rebonato for the previous 12 months.
Richard White holds a doctorate in Particle Physics from Imperial College London, and a firstclass honours diploma in Physics from Oxford University. He held a Research Associate place at Imperial College earlier than becoming a member of RBS in 2004 as a Quantitative Analyst. His analysis pursuits embrace choice pricing with Levy Processes, Genetic Algorithms for portfolio optimisation, and Libor Market Models with stochastic volatility. He is at the moment taking a fortuitously timed sabbatical to pursue his joint ardour for journey and scuba diving.