Jacobi stochastic volatility factor for the LIBOR market model

Abstract

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 Hestontype 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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“…, the instantaneous correlation is smaller than ρ(t) at each time. Observing that Q(v) → v when (v min , v max ) → (0, +∞), it can be shown that the swap rate process defined by (3.4) converges weakly (and strongly) towards the process defined in (3.3) (see [8]); thus, present framework (3.4) can be seen as a numerical approximation of (3.3). The fact that the volatility process remains bounded through time allows to write swaptions prices as convergent series whose coefficient are linear combinations of the moments of the swap rates.…”

“…It has been introduced in [8] the Jacobi version of the DDSVLMM which is an approximation of the model (3.3). In the proposed setting, the volatility factor is modelled by a [v min , v max ]-valued Jacobi process.…”