A Comparison of Biased Simulation Schemes for Stochastic Volatility Models

Tinbergen Institute Discussion Paper No. 06-046/4

30 Pages Posted: 19 May 2006 Last revised: 20 Mar 2008

See all articles by Roger Lord

Roger Lord

Cardano Risk Management

Remmert Koekkoek

Credit Suisse

Dick J. C. van Dijk

Erasmus University Rotterdam - Erasmus School of Economics - Econometric Institute; ERIM

Date Written: February 6, 2008

Abstract

Using an Euler discretisation to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and J├Ąckel and Ninomiya and Victoir, as well as to the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme outperforms all considered biased schemes in terms of bias and root-mean-squared error.

Keywords: Stochastic volatility, Heston, square root process, CEV process, Euler-Maruyama, discretisation, strong convergence, weak convergence, boundary behaviour

JEL Classification: C63, G13

Suggested Citation

Lord, Roger and Koekkoek, Remmert and van Dijk, Dick J.C., A Comparison of Biased Simulation Schemes for Stochastic Volatility Models (February 6, 2008). Tinbergen Institute Discussion Paper No. 06-046/4. Available at SSRN: https://ssrn.com/abstract=903116 or http://dx.doi.org/10.2139/ssrn.903116

Roger Lord (Contact Author)

Cardano Risk Management ( email )

Rotterdam 3011 AA
Netherlands

Remmert Koekkoek

Credit Suisse ( email )

CRTI 4
P.O. Box
Zurich, CH-8070
Switzerland

Dick J.C. Van Dijk

Erasmus University Rotterdam - Erasmus School of Economics - Econometric Institute

P.O. Box 1738
3000 DR Rotterdam
Netherlands

ERIM ( email )

P.O. Box 1738
3000 DR Rotterdam
Netherlands
+31 10 408 1263 (Phone)
+31 10 4089162 (Fax)

HOME PAGE: http://people.few.eur.nl/djvandijk

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