Inria / Raweb 2004
Team: Mathfi

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Team : mathfi

Section: New Results

Keywords: quantization.

Quantization methods

Participants: G. Pagès, J. Printems.

The quantization method applied to mathematical finance or more generally to systems of coupled stochastic differentials equations (Forward/Backward) as introduced in [52] consists in the approximation of the solution of the Backward Kolmogorov equation by means of piecewise constant functions defined on an appropriate Voronoï tesselation of the state space (Rd). The numerical aspects of such a method are to compute such tesselations adapted to the underlying diffusion and to estimate theirs transition probabilities between different cells of two successive meshes (after a time discretization procedure). Hence, it allows the computation of a great number of conditional expectations along the diffusions paths.

For these reasons, such a method seems to work well with the problem of valuation and hedging of financial products. More generally, its applications are concerned with the American options [52],[2], [16], stochastic control [25], nonlinear filtering and related problems (Zakai and McKean-Vlasov type stochastic partial differential equation [67]). See also [26] for a review on the subject.

There exists an ``infinite dimensional'' version of the quantization method. In particular, when a stochastic process is viewed as an Hilbert value random variable. It is the purpose of the functional quantization to study such quantization (see [78]). This new point of view allows us to treat the problem of valuation of Asiatic options or more generally of ``path dependant'' options in an L2 framework. See [32] for a numerical study for Asiatic options and European options in the Heston stochastic volatility model. In this paper, log-Romberg extrapolation in space seems to provide very good results. A theoretical justification of such numerical results should be a promising field of study. Note that this should be implemented in the software premia (release 8) (see Premia).

As concerned nonlinear filtering and quantization, let us note a different way of study. In a work in collaboration with Bruno Saussereau (Université de Franche-Comté, Besançon, France), we study the filtering of nonlinear systems from a numerical point of view: we want to compute the conditional expectations of signals when the observation process and/or the dynamic of the signal is not linear by means of a spectral approximation of the Zakai equation. In cite [77], the authors have proposed a spectral approach of nonlinear filtering by means of the Chaos expansion of the Wiener process. Numerical experiments based on this approach together with a quantization method provide promising results (see [40]).