Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 359142, 17 pages
Research Article

A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility

Elisa Alòs,1 Jorge A. León,2 Monique Pontier,3 and Josep Vives4

1Departament d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain
2Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), Apartado Postal 14-740, CP 07000 México D.F., Mexico
3Institut Mathématique de Toulouse, Université de Toulouse, 31062 Toulouse cedex 9, France
4Departament de Probabilitat, Lògica i Estadística, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Received 1 April 2008; Revised 3 September 2008; Accepted 25 November 2008

Academic Editor: Wenbo Li

Copyright © 2008 Elisa Alòs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We obtain a Hull and White type formula for a general jump-diffusion stochastic volatility model, where the involved stochastic volatility process is correlated not only with the Brownian motion driving the asset price but also with the asset price jumps. Towards this end, we establish an anticipative Itô's formula, using Malliavin calculus techniques for Lévy processes on the canonical space. As an application, we show that the dependence of the volatility process on the asset price jumps has no effect on the short-time behavior of the at-the-money implied volatility skew.