Jacobi Stochastic Volatility factor for the Libor Market Model
Résumé
We propose a new method to efficiently price swap rates derivatives
under the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion (DDSVLMM).
This method uses polynomial processes combined with Gram-Charlier expansion techniques. The
standard pricing method for this model relies on dynamics freezing to recover an Heston-type
model for which analytical formulas are available. This approach is time consuming and efficient
approximations based on Gram-Charlier expansions have been recently proposed. In this article,
we first discuss the fact that for a class of stochastic volatility model, including the Heston
one, the classical sufficient condition ensuring the convergence of the Gram-Charlier series can
not be satisfied. Then, we propose an approximating model based on Jacobi process for which we can
prove the stability of the Gram-Charlier expansion. For this approximation, we have been able to
prove a strong convergence toward the original model; moreover, we give an estimate of the
convergence rate. We also prove a new result on the convergence of the Gram-Charlier series when
the volatility factor is not bounded from below. We finally illustrate our convergence results with
numerical examples.
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