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

Shaw, WT; Schofield, M; (2012) A model of returns for the post-credit-crunch reality: hybrid Brownian motion with price feedback. Quantitative Finance 10.1080/14697688.2011.642810.

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Recent market events 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 trade arrival model with variable size orders and a general arrival-time distribution. With simplifications we are led in the jump-free case to a local volatility model defined by a hybrid SDE mixing both arithmetic and geometric or CIR Brownian motions, whose solution in the geometric case 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 time-dependent Johnson-SU distributions and 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 behaviors. 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. We touch briefly on the extent to which such a model is consistent with the dynamics of a ‘flash-crash’ event, and briefly explore the statistical evidence for our model.

Type: Article
Title: A model of returns for the post-credit-crunch reality: hybrid Brownian motion with price feedback
Location: UK
DOI: 10.1080/14697688.2011.642810
Publisher version: http://www.tandfonline.com/doi/abs/10.1080/1469768...
UCL classification: UCL > School of BEAMS
UCL > School of BEAMS > Faculty of Maths and Physical Sciences
URI: http://discovery.ucl.ac.uk/id/eprint/1336795
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