## Section: New Results

### A multifractional Hull and White model

Participants : Joachim Lebovits, Jacques Lévy Véhel.

*In collaboration with Sylvain Corlay (Paris 6 University).*

We have considered the following model, which is an extension of the fractional Hull and White model proposed in [55] : under the risk-neutral measure, the forward price of a risky asset is the solution of the S.D.E.

where ${B}_{t}^{h}$ is a multifractional Brownian motion with regularity function $h$, and ${W}_{t},{W}_{t}^{\sigma}$ are standard Brownian motions. This SDE is interpgreted in the Wick-Itô sense.

Using functional quantization techniques, it is possible to compute numerically implied forward start volatilities for this model. Using an adequate $h$ function estimated from SP500 data, we have shown that this model is able to reproduce to some extent the volatility surface observed on the market [34] .