## Section: New Results

Keywords : quantization.

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

Participants : G. Pagès, J. Printems, A. Sellami.

G. Pagès and A. Sellami have carried out research on theoretical aspects of optimal Quantization, in both finite and infinite dimesnional settings.

In R^{d} , G. Pagès solved with S. Graf and H. Luschgy the so-called (r, s) -problem i.e. we elucidated the asymptotic behaviour of a sequence of L^{r} -optimal quantizers when used as quantizers in L^{s} . We gave some lower bound or this behaviour and gave sufficient conditions to ensure that they preserve the standard rate of quantization ie . This has many applications for higher order cubature formula for numerical integration and conditional expectation approximation.

With H. Luschgy, G. Pagès pointed out the strong connection between pathwise L^{p} regularity of processes and the -quantization rate for the pathwise L^{r}([0, T], dt) -norm of these processes. We apply these very general connection to Lévy processes where we showed that grosso modo, the quantization rate of a Lévy process is ruled by where (X) denotes the Blumenthal-Getoor index of the Lévy process. We also provided some rates for the compound Poisson processes.

The above results (as concerns Lévy processes) rely on an extension of an old resulst by Millar about the (absolute) moments of Lévy processes E|X_{t}|^{r} as t0 .

G. Pagès and A. Sellami have made a connection between rough paths and functional quantization with some applications to the pricing of European pth-dependent options.

G. Pagès and J. Printems have launched a large scale computation of optimized quantization grids (from d = 1 up to d = 10 ) which improves the former one. They also completed a large scale computation of(quadratic) optimal functional quantization grids for the Brownian motion (from N = 1 up to N = 10000 ). Some of them are available on the website devoted to quantization http://www.quantize.maths-fi.com/ G. Pagès and J. Printems developped a pricer of Asian options in the Heston model based on a optimal functional quantization, included in the 2006 issue of Premia.

G. Pagès and A. Sellami are currently working on the pricing of swing option on commodities : theoretical aspects and numerical methods (optimal quantization).

G. Pagès is also working on multistep Romberg extrapolation in the Monte Carlo method in presence of an expansion of the time discretization error, with application to exotic path-dependent otpions. With his student F. Panloup, he is studying a recursive algorithm for the computation of functional of a stationary jump diffusion.