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Malliavin differentiability of the Heston volatility and applications to option pricing

Part of: Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Elisa Alòs  and
Christian-Oliver Ewald
Elisa Alòs*
Affiliation:
University Pompeu Fabra Barcelona
Christian-Oliver Ewald*
Affiliation:
University of St Andrews
*
∗ Postal address: Department of Economics, University Pompeu Fabra Barcelona, c/Ramon Trias fargas, 25-27, 08005 Barcelona, Spain. Email address: elisa.alos@upf.edu
∗∗ Postal address: Department of Economics, University of St Andrews, St Salvator's College, St Andrews, Fife KY16 9AL, UK. Email address: c.o.ewald@st-andrews.ac.uk
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Abstract

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We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of Alòs (2006) in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.

Keywords

Malliavin calculusstochastic volatility modelHeston modelCox-Ingersoll-Ross processHull and White formulaoption pricing

MSC classification

Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 144 - 162
Copyright
Copyright © Applied Probability Trust 2008 

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