A model of returns for the post-credit-crunch reality: Hybrid Brownian motion with price feedback

The market events of 2007-2009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the medium-term log-return dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival model with variable size orders. With simplifications we are led to a hybrid SDE mixing both arithmetic and geometric Brownian motions, whose solution is given by a class of integrals of exponentials of one Brownian motion against another, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of “Pearson diffusion”, or equivalently a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under mean-reverting circumstances we arrive naturally at an equilibrium fat-tailed return distribution with a Student or Pearson Type IV form. Under less restrictive assumptions richer dynamics are possible, including bimodal structures. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that “25σ” events are significantly more probable. We exhibit simple example solutions of the Fokker-Planck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviours. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity.