On Valuing Constant Maturity Swap Spread Derivatives

Leonard Tchuindjo

US Department of the Treasury, Washington DC, USA.

DOI: 10.4236/jmf.2012.22020                

Abstract

Motivated by statistical tests on historical data that confirm the normal distribution assumption on the spreads between major constant maturity swap (CMS) indexes, we propose an easy-to-implement two-factor model for valuing CMS spread link instruments, in which each forward CMS spread rate is modeled as a Gaussian process under its relevant measure, and is related to the lognormal martingale process of a corresponding maturity forward LIBOR rate through a Brownian motion. An illustrating example is provided. Closed-form solutions for CMS spread options are derived.

Keywords

CMS Spread; Market Model; Brownian Motion; Forward Measure

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Fannie Mae, “Fannie Mae Universal Debt Facility,” Fed- eral Home Loan Association, Washington DC, 2008. http://www.fanniemae.com/markets/debt/pdf/CUSIP_PS31398ANE8.pdf
[2]	R. Carmona and V. Durrleman, “Pricing and Hedging of Spread Options,” SIAM Review, Vol. 45, No. 4, 2003, pp. 627-685. doi:10.1137/S0036144503424798
[3]	D. Belomestny, A. Kolodko and J. Schoenmakers, “Pricing Spreads in the Libor Market Model,” Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2008.
[4]	M. Lutz and R. Kiesel, “Efficient Pricing of CMS Spread Options in a Stochastic Volatility LMM,” Institute of Mathematical Finance, Ulm University, Ulm, 2010.
[5]	W. Margrabe, “The Value of an Option to Exchange One Asset for the Another,” Journal of Finance, Vol. 33, No. 1, 1978, pp. 177-186. doi:10.2307/2326358
[6]	H. Geman, N. Karoui and J. C. Rochet, “Change of Numeraire, Change of Probability Measure and Option Pricing,” Journal of Applied Probability, Vol. 32, No. 2, 1995, pp. 443-458. doi:10.2307/3215299
[7]	F. Jamshidian, “Bond and Option Evaluation in the Gaussian Interest Rate Model,” Research in Finance, Vol. 9, 1991, pp. 131-170.
[8]	A. V. Antonov and M. Arneguy, “Analytical Formulas for Pricing CMS Products in the Libor Market Model with Stochastic Volatility,” Numerix Software Ltd., London, 2009.
[9]	A. Pelsser, “Efficient Methods for Valuing Interest Rate Derivative,” Springer-Verlag, New York, 2000.