Eur. Phys. J. B (2012) 85: 276
https://doi.org/10.1140/epjb/e2012-30134-y

Regular Article

Baldovin-Stella stochastic volatility process and Wiener process mixtures

¹ Capital Fund Management, 6-8 boulevard Hausmann, 75009 Paris, France
² Chaire de finance quantitative, Laboratoire de mathématiques appliquées aux systèmes, École Centrale Paris, Grande Voie des Vignes, 92290 Châtenay-Malabry, France
³ Encelade Capital SA, ch. du Bochet 8a, 1025 St-Sulpice, Switzerland

^a e-mail: damien.challet@ecp.fr

Received: 14 February 2012
Published online: 6 August 2012

Abstract

Starting from inhomogeneous time scaling and linear decorrelation between successive price returns, Baldovin and Stella recently proposed a powerful and consistent way to build a model describing the time evolution of a financial index. We first make it fully explicit by using Student distributions instead of power law-truncated Lévy distributions and show that the analytic tractability of the model extends to the larger class of symmetric generalized hyperbolic distributions and provide a full computation of their multivariate characteristic functions; more generally, we show that the stochastic processes arising in this framework are representable as mixtures of Wiener processes. The basic Baldovin and Stella model, while mimicking well volatility relaxation phenomena such as the Omori law, fails to reproduce other stylized facts such as the leverage effect or some time reversal asymmetries. We discuss how to modify the dynamics of this process in order to reproduce real data more accurately.