Skewed Libor Market Model and Gaussian HJM explicit approaches to rolled deposit options

Henrard, Marc (2007): Skewed Libor Market Model and Gaussian HJM explicit approaches to rolled deposit options.

Abstract

A simple exotic option (floor on rolled deposit) is studied in the shifted log-normal Libor Market (LMM) and Gaussian HJM models. The shifted log-normal LMM exhibits a controllable volatility skew. An explicit approach is used for both models. Using approximations the price in the LMM is obtained without Monte Carlo simulation. The more precise approximation uses a twisted version of the perdictor-corrector adapted to explicit solutions. The results of the approximation are surprisingly good.

• Item Type: MPRA Paper
• Institution: Bank for International Settlements
• Language: English
• Keywords: Libor Market Model; Heath-Jarrow-Morton; skew; smile; explicit solution; approximation; Bond Market Model; option on composition; existence results
• Subjects: G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing; E - Macroeconomics and Monetary Economics > E4 - Money and Interest Rates > E43 - Interest Rates: Determination, Term Structure, and Effects; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling

References

R. Baviera. Bond market model. International Journal of Theoretical and Applied Finance, 9(4):577--596, 2006.

A. Brace, D. Gatarek, and M. Musiela. The market model of interest rate dynamics. Mathematical Finance, 7:127--154, 1997.

D. C. Brody and L. P. Hughston. Chaos and coherence: a new framework for interest-rate modelling. Proc. R. Soc. Lond. A., 460:85--110, 2004.

A. Daniluk and D. Gatarek. A fully log-normal Libor market model. Risk, 18(9):115--118, September 2005.

E. Errais and F. Mercurio. Yes, libor models can capture interest rate derivatives skew: a simple modelling approach. Working paper 680621, SSRN, February 2005. Available at SSRN: http://ssrn.com/abstract=680621.

D. Gatarek. Nonparametric calibration of forward rate models. Technical report, NumeriX, 2005.

P. Hagan, D. Kumar, A. Lesniewski, and D. Woodward. Managing smile risk. Wilmott Magazine, Sep:84--108, 2002.

D. Heath, R. Jarrow, and A. Morton. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60(1):77--105, January 1992.

M. Henrard. Libor Market Model and Gaussian HJM explicit approaches to option on composition. Working Paper 888484, SSRN, November 2005. Available at http://ssrn.com/abstract=888484.

M. Henrard. A semi-explicit approach to Canary swaptions in HJM one-factor model. Applied Mathematical Finance, 13(1):1--18, March 2006.

T. S. Y. Ho and S.-B. Lee. Term structure movements and pricing of interest rate contingent claims. Journal of Finance, 41:1011--1029, 1986.

P. J. Hunt and J. E. Kennedy. Financial Derivatives in Theory and Practice. Wiley series in probability and statistics. Wiley, second edition, 2004.

C. Hunter, P. Jackel, and M. Joshi. Getting the drift. Risk, July 2001.

F. Jamshidian. Libor and swap market models and measures. Finance and Stochastics, 1:295--330, 1997.

M. Joshi and A. Stacey. New and robust drift approximation for the Libor Market Model. Working Paper 907385, SSRN, February 2006.

P. Kloden and E. Platen. Numerical Solution of Stochastic Differential Equations. Spinger, Berlin, Heidelberg, New-York, 1995.

K. Miltersen, K. Sandmann, and D. Sondermann. Closed form solution for term structure derivatives with lognormal interest rates. Journal of Finance, 52(1):409--430, 1997.

L. T. Nielsen. Pricing and hedging of derivative securities. Oxford University Press, 1999.

R. Rebonato. Modern pricing of interest-rate derivatives: the LIBOR market model and Beyond. Princeton University Press, Princeton and Oxford, 2002.