High order discretization schemes for stochastic volatility models
AbstractIn 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 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, 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].
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Date of creation: 07 Aug 2009
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discretization schemes; stochastic volatility models; weak trajectorial convergence; multilevel Monte Carlo;
This paper has been announced in the following NEP Reports:
- NEP-ALL-2009-08-22 (All new papers)
- NEP-ALL-2010-06-11 (All new papers)
- NEP-CMP-2009-08-22 (Computational Economics)
- NEP-ETS-2009-08-22 (Econometric Time Series)
- NEP-ETS-2010-06-11 (Econometric Time Series)
- NEP-MST-2009-08-22 (Market Microstructure)
- NEP-ORE-2009-08-22 (Operations Research)
- NEP-ORE-2010-06-11 (Operations Research)
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