High order discretization schemes for stochastic volatility models

In typical stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose a scheme, based on the Milstein discretization of this SDE, which converges with order one to the asset price dynamics for an appropriate notion of convergence that we call weak trajectorial convergence, a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence to the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Ornstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles (Multilevel Monte Carlo path simulation. Operations Research, 56:607-617, 2008).