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Analytical Formulas for Pricing CMS Products in the LIBOR Market Model with the Stochastic Volatility

32 Pages Posted: 10 Mar 2009  

Date Written: March 3, 2009

Abstract

In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. The derived formulas can also be used for model calibration to the market, including European swaptions and CMS products.

The first technical achievement of this work is related to the optimal calculation of the measure change. For single-rate CMS products, we have used the standard linear regression of the measure change, with optimally calculated coefficients. For the CMS spread options, where the linear procedure does not work, we propose a new effective 'non-linear' measure change technique. The fit quality of the new results is confirmed numerically using Monte Carlo simulations.

The second technical advance of the article is a theoretical derivation of the generalized spread option price via two-dimensional Laplace transform presented in a closed form in terms of the complex Gamma-functions.

Keywords: LMM, stochstic volatility, CMS swaps, CMS caps, CMS spread option, Markovian Projection

JEL Classification: C1, C3, C5, C6

Suggested Citation

Antonov, Alexandre and Arneguy, Matthieu, Analytical Formulas for Pricing CMS Products in the LIBOR Market Model with the Stochastic Volatility (March 3, 2009). Available at SSRN: https://ssrn.com/abstract=1352606 or http://dx.doi.org/10.2139/ssrn.1352606

Alexandre Antonov (Contact Author)

Numerix ( email )

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Matthieu Arneguy

Numerix ( email )

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