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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.

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

Abstract:
Markovian Projection is an optimal approximation of a complex underlying process with a simpler one, keeping essential properties of the initial process. The Heston process, as the Markovian Projection target, is an example.

In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. As an important application, we derive an effective approximation for FX/EQ options for the Heston model, coupled with correlated Gaussian interest rates.

The main technical result is an option evaluation for correlated Heston/Lognormal processes. Unlike the case of exactly solvable (affine) zero correlation or its uncorrelated displacement generalization,considered by Andreasen, non-trivial correlations destroy affine structure and exact solvability.

Using the powerful technique of Markovian Projection onto a Heston model with displaced volatility, we produce an effective approximation and present its numerical confirmation.

Markovian projection, stochastic volatility, Heston model, Gyongy lemma, Heston/Hull-White correlated hybrid, FX-options approximation