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High order discretization schemes for stochastic volatility models

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  • Benjamin Jourdain

    (CERMICS)

  • Mohamed Sbai

    (CERMICS)

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].

Suggested Citation

  • Benjamin Jourdain & Mohamed Sbai, 2009. "High order discretization schemes for stochastic volatility models," Papers 0908.1926, arXiv.org, revised Oct 2011.
  • Handle: RePEc:arx:papers:0908.1926
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    File URL: http://arxiv.org/pdf/0908.1926
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    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
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