后信贷危机现实的收益模型：价格反馈的混合布朗运动
A model of returns for the post-credit-crunch reality: Hybrid Brownian motion with price feedback
摘要：
2007 - 2009年的市场事件重新激发了人们对现实回报模型的探索，这些模型能够捕捉到极端波动的更大可能性。在本文中，我们建立了一个市场的中期日志回报动态的基本面和技术的交易者。这是基于一个泊松贸易到达模型与可变大小的订单。通过简化，我们得到了算术布朗运动和几何布朗运动的混合解，其解由一类布朗运动的指数积分给出，其形式由Yor和合作者考虑。将混合型扩散简化为单一的布朗运动导致Nagahara所考虑的形式的扩散，这是一种"皮尔逊扩散"，或者等价地是双曲型扩散。根据交易的平衡，各种动态和均衡都是可能的。在均值回复的情况下，我们自然地得出一个均衡的厚尾收益分布，它具有学生或皮尔逊IV型形式。在较少限制的假设下，更丰富的动态是可能的，包括双峰结构。方差爆炸的现象被确定，引起更大的价格变动，可能有一个先验的预期，使"$25σ$"事件的可能性更大。我们展示了福克-普朗克方程的简单示例解，展示了这种方差爆炸如何隐藏在标准高斯立面之下。这些是一个扩展的分布类的基本成员，具有丰富多样的结构，能够描述广泛的市场行为。几种方法的密度函数是可能的，并给出了一个例子的计算的双曲风险值。该模型还建议推广的Bougerol身份。
英文摘要：
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.
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