## Section: New Results

### Option Pricing and Calibration

#### Calibration of regime-switching local volatility models

Participant : Benjamin Jourdain.

By Gyongy's theorem, a local and stochastic volatility model is calibrated to the market prices of all call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labordère (2011), provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, B. Jourdain and A. Zhou prove existence to the associated Fokker-Planck equation. Thanks to Figalli (2008), they then deduce existence of a new class of fake Brownian motions. They extend these results to the special case of the LSV model called Regime Switching Local Volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, they prove existence to the associated Fokker-Planck PDE. They finally deduce existence of the calibrated model by extending the results in Figalli (2008).

#### American options

Participant : Damien Lamberton.

With Mihail Zervos, D. Lamberton has worked on American options involving the maximum of the underlying asset. With Giulia Terenzi, he has been working on American options in Heston's model. They obtained results about existence and uniqueness for the associated variational inequality, in suitable weighted Sobolev spaces (see Feehan and co-authors for recent results on elliptic problems).