Share Price Movements in the Post-Credit-Crunch environment

The market events of 2007-2008 have reinvigorated the search for realistic share price models that capture greater likelihoods of extreme movements and that also reflect fear- and greed-linked trading expressed through price-motivated trades. In this paper we model the medium-term log-return dynamics in a market containing both fundamental and technical traders. This is done in a simple way based on a Poisson trade arrival model with variable size orders. With simplifying assumptions we are led to a novel SDE mixing arithmetical and geometric Brownian motions on the log-return. 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 $t$ or skew-Student (Pearson Type IV) form. Under less restrictive assumptions still richer dynamics are possible. One special case leads to a natural hyperbolic variation of the OU SDE. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that "25 sigma" events can become more commonplace. We exhibit a solution of the Fokker-Planck equation for a special case that shows how such variance explosion can hide beneath a standard Gaussian facade.