The Ups and Downs of Modeling Financial Time Series with Wiener Process Mixtures

TL;DR

This paper refines a financial time series model based on Wiener process mixtures, replacing distributions for better analytical tractability and discussing modifications to capture more market phenomena.

Contribution

It explicitly formulates a model using Student and generalized hyperbolic distributions, extending its analytical tractability and discussing how to improve its realism.

Findings

Model can be expressed as mixtures of Wiener processes.

It captures volatility relaxation phenomena like the Omori law.

Fails to reproduce leverage effect and some asymmetries.

Abstract

Starting from inhomogeneous time scaling and linear decorrelation between successive price returns, Baldovin and Stella recently proposed a 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 L\'evy distributions; we also 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, the stochastic processes arising in this framework are representable as mixtures of Wiener processes. The 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.