High order discretization schemes for stochastic volatility models

TL;DR

This paper introduces high-order discretization schemes for stochastic volatility models, achieving improved convergence rates and demonstrating their effectiveness through numerical experiments and compatibility with multilevel Monte Carlo methods.

Contribution

It proposes new discretization schemes based on Milstein and Ninomiya-Victoir methods with higher weak convergence orders for stochastic volatility models.

Findings

Schemes achieve order one and two weak convergence for asset prices.

Numerical experiments confirm theoretical convergence rates.

Schemes are well-suited for multilevel Monte Carlo methods.

Abstract

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].