High order discretization schemes for stochastic volatility models

Benjamin Jourdain and Mohamed Sbai
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Benjamin Jourdain: CERMICS
Mohamed Sbai: CERMICS

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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].

Date: 2009-08, Revised 2011-10
New Economics Papers: this item is included in nep-ets
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:0908.1926

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