RICARDO REBONATO, KENNETH MCKAY, RICHARD WHITE – THE SABR-LIBOR MARKET MODEL

A major innovation in the interest rate space is presented in this book. It explains a financially motivated extension of the LIBOR Market model which accurately reproduces the prices for plain vanilla hedging instruments ( swaps and caplets) of all strikes and maturities produced by the SABR model. The authors show how to recover the entire smile surface using their extension of the LIBOR market model. This is not just a new model, this is a new way of option pricing that takes into account the need to calibrate as accurately as possible to the plain vanilla reference hedging instruments and the need to obtain prices and hedges in reasonable time whilst reproducing a realistic future evolution of the smile surface. The framework that the authors provide reproduces today’s market prices of plain vanilla options almost exactly and simultaneously gives a reasonable future evolution for the smile surface removes the hard choice between accuracy and time.

The SABR model is a good model for European options and is the starting point for the extension of the LMM. The problem with SABR is that it treats each European option in isolation and the processes for the various underlyings do not talk to each other so it isn’t obvious how to relate these processes into the dynamics of the whole yield curve. The authors brought the dynamics of the various forward rates and stochastic volatilities under a single umbrella with this new model. They derive drift adjustments to be applied to both the forward rates and their volatilities. Complex derivatives that depend on the joint realization of all relevant forward rates can now be priced.

CONTENTS

The geographic set-up. The market model of libor. The SABR is a model. The LMM-SABR is a model.

IMPLEMENTATION AND CALIBRATION

The LMM-SABR model is Calibrated to Market Caplet prices. The LMM/SABR model is Calibrated. The SABR-LIBOR Market Model is Calibrating the Correlation Structure.

EMPIRICAL EVIDENCE

The problem of Empirical. Estimating the forward rates. Estimating the correlation structure. Estimating the volatility.

HEDGING

Hedging the volatility structure is something that needs to be done. Hedging the Correlation structure is difficult. Hedging in market stress.

TABLE OF CONTENTS

1. INTRODUCTION.

2. THE THEORETICAL SET-UP.
3. The LIBOR Market Model.

There are 2.1 definitions.

The function of the volatility.

The Correlation is separated from the Volatility Term.

There is a Caplet-Pricing Condition again.

There is a correlation between the forward-rate and the forward-rate.

There are possible shapes of the douth correlation function.

The Covariance Integral Again.

3. THE SABR MODEL.

The SABR model is a good model.

The model has a description.

The option prices are given by the SABR model.

There are 3.4 special cases.

The SABR model has a Qualitative Behaviour.

The link between the exponent and the volatility.

There is volatility clustering in the model.

The market. Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

How do we know that the market has chosen?

There are problems with the SABR model.

4. The LMM-SABR Model.

The equations of motion are used.

Our model introduced the nature of the Stochasticity.

A simple correlation structure.

There is a more general correlation structure.

There are 4.5 observations on the correlation structure.

There is a volatility structure.

What do we mean by time Homogeneity?

There is a volatility in periods of market stress.

There is a more general Stochastic Volatility Dynamics. Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

The No-Arbitrage Drifts are calculated.

1. IMPLEMENTATION AND CALIBRATION.

5 CALIBRATING THE LMM-SABR MODEL TO MARKET CAPLET PRICES.

The Caplet-Calibration Problem is a problem.

The initial and the function’s parameters are chosen.

Values, kT 0.

The Parameters of the Function h.

The choice of the exponent and the correlation.

There were 5.5 results.

Calibration in practice has implications for the SABR model.

There are implications for model choice.

6. CALIBRATING THE LMM-SABR MODEL TO MARKET SWAPTION PRICES.

There is a swaption Calibration Problem.

There are swaps and forward rate dynamics.

The Instantaneous Swap Rate Volatility is Approximating.

Approximating the initial value of the swap rate.

Vimating the second route and the volatility of the swap rate.

Approximating the swap rate and volatility correlation.

Approximating the swap rate exponent.

There were 6.8 results.

Suggestions for future work. Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

Derivation of Approximate Swap Rate Volatility is included in the Appendix.

Derivation of swap rate/ swap rate volatility correlation

Approximation of.

7. CALIBRATING THE CORRELATION STRUCTURE.

There is a statement of the problem.

A model matrix is created.

Calibration using the Hypersphere Method is a case study.

Which method is best for you?

Appendix 1.

III. EMPIRICAL EVIDENCE. RICARDO REBONATO, KENNETH MCKAY, RICHARD WHITE – THE SABR-LIBOR MARKET MODEL

8. THE EMPIRICAL PROBLEM.

There is a statement about the Empirical Problem.

What do we know about the literature?

There is a data description.

The distributional analysis has its limitations.

What is the true exponent?

Appendix: Some Analytic Results.

9. ESTIMATING THE VOLATILITY OF THE FORWARD RATES.

Expiry-Dependence of Forward Rates.

Direct Estimation.

The normality of the residuals is looked at.

There are variations on the theme.

There is information about the options market.

Overall Conclusions.

10. ESTIMATING THE CORRELATION STRUCTURE.

We are trying to do something.

Results from Random Matrix Theory.

There is an empirical estimation. Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

There areDescriptive Statistics.

There are Empirical Correlation Blocks.

What does Random Matrix Theory tell us?

The Correlation Matrices are Calibrating.

What amount of information do the proposed models retain?

1. HEDGING.
2. Various Types of Hedging.

There is a statement of the problem.

There are three types of hedging.

There are definitions.

The first-order derivatives have respect to the underlyings.

The second-order derivatives have respect to the underlyings.

There is a generalizing of Functional-Dependence Hedging.

How does the model know about Vanna and Volga? Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

There is a choice of hedging instrument.

12. HEDGING AGAINST MOVES IN THE FORWARD RATE AND IN THE VOLATILITY.

The SABR-(LMM) Model has Delta Hedging in it.

The SABR-(LMM) Model has a Vega Hedging in it.

13. (LMM)-SABR HEDGING IN PRACTICE: EVIDENCE FROM MARKET DATA.

This chapter has a purpose.

Notation.

There are results for the SABR model.

The LMM-SABR model has hedging results.

Conclusions.

14. HEDGING THE CORRELATION STRUCTURE.

The problem has Intuition behind it.

The forward-rate block is being Hedging.

The volatility-rate block is being hedged.

The forward-rate/volatility block is being Hedging.

There are final considerations.

15. HEDGING IN CONDITIONS OF MARKET STRESS.

There is a statement of the problem.

The function of the volatility. Richard White and Kenneth McKay are part of the SABR-LIBOR Market Model.

There is a case study.

Hedging.

There were 15 results.

Are we getting anything for nothing?

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