  In usual 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 It\^o'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, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya-Victoir
discretization of this SDE, with order two of weak convergence for the asset
price. We also propose a specific scheme with improved convergence properties
when the volatility of the asset price is driven by an Orstein-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 [2008a, 2008b].
