## Section: New Results

Keywords : quantization.

### Functional quantization for option pricing in a non Markovian setting

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
[64] 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
R^{d} . The numerical procedure consists in computing such tesselations adapted to the underlying diffusion and estimating the 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 is efficient for pricing and hedging financial products. More generally, it can be applied to American options [64] , [65] , [24] , stochastic control [100] , nonlinear filtering and related problems (Zakai and McKean-Vlasov type stochastic partial differential equation [85] ). See also [101] for a review on the subject.

The aim of optimal quantization is to study the best
L^{2} approximation of Hilbert valued random variables taking at most
Nvalues. This approach allows us to study the numerical quantization of the Brownian motion from a functional point of view by considering the Brownian motion as a random variable taking values in
L^{2}([0,
T]) . Similar approach can be considered for other Gaussian and non Gaussian processes. Unfortunatly, the resulting quantization error has a bad rate with respect to
N, namely
1/log(
N) . Nevertheless, numerical computations tell us that things behave better than expected.

In a financial framework, functional quantization is helpful when dealing with options with ``non Markovian'' payoffs, that is payoffs depending on the whole trajectory of the asset price process, such as time average (Asian) options or maximum (lookback) options. Functional quantization
is also useful in the case of ``Markovian'' payoffs in a stochastic volatility framework since the values of the asset at the maturity may depend on the whole path of the volatility. In these cases, we can approximate the value of the option by the usual numerical integration in a functional
space considering the asset price process as some
H-valued random variable rather than as a
R^{d} -valued process. Numerical study in the framework of the Heston model can be found in
[41] .