ABSTRACT
The calibration of local stochastic volatility models to market smiles leads to McKean nonlinear stochastic differential equations (SDEs), introduced in Chapter 10. As shown in Section 10.1, the Fokker-Planck equations associated to McKean SDEs are nonlinear. In this chapter, we review various methods for solving such nonlinear PDEs. For one-factor stochastic volatility models, we can rely on finite difference schemes. But, as PDEs suffer from the curse of dimensionality, we must turn to probabilistic methods to handle multi-factor stochastic volatility models. In this chapter, we present several such Monte Carlo methods, such as the particle algorithm introduced in Section 10.2. We also devote ample space to the case of stochastic interest rates, which can be easily handled by the particle method. Parts of this research have been published in [128, 122].
