High order discretization schemes for stochastic volatility models

Benjamin Jourdain () and Mohamed Sbai ()
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Benjamin Jourdain: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École nationale des ponts et chaussées, MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées
Mohamed Sbai: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École nationale des ponts et chaussées

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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 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].

Keywords: discretization schemes; stochastic volatility models; weak trajectorial convergence; multilevel Monte Carlo (search for similar items in EconPapers)
Date: 2013
Note: View the original document on HAL open archive server: https://hal.science/hal-00409861v4
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Citations: View citations in EconPapers (2)

Published in The Journal of Computational Finance, 2013, 17 (2), pp.113-165. ⟨10.21314/JCF.2013.262⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00409861

DOI: 10.21314/JCF.2013.262

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