摘要翻译：
2007-2009年的市场事件重新激发了人们寻找现实回报模型的热情，这些模型捕捉到了极端波动的更大可能性。在本文中，我们建立了一个市场中的中期对数收益动态模型，市场中同时有基本面和技术性交易者。这是基于一个泊松贸易到达模型与可变规模的订单。通过简化，我们得到了一个混合的SDE混合算术布朗运动和几何布朗运动，它的解是由一个布朗运动对另一个布朗运动的指数积分给出的，其形式是Yor和合作者所考虑的。将混合SDE简化为单个布朗运动，得到了Nagahara所考虑的形式的SDE，这是一种“皮尔逊扩散”，或等效为双曲OU SDE。各种动态和均衡是可能的，取决于交易的平衡。在均值回归的情况下，我们自然地到达一个学生型或皮尔逊型IV型的平衡胖尾回报分布。在限制较少的假设下，包括双峰结构在内的更丰富的动力学是可能的。方差爆炸的现象被识别出来，它会引起更大的价格波动，而这可能是先验预期的，因此“$25\sigma$”事件的可能性明显更大。我们展示了福克-普朗克方程的简单示例解，说明了这种方差爆炸是如何隐藏在标准高斯表面之下的。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。文中给出了计算双曲型VaR的一个实例。该模型还建议了Bougerol恒等式的推广。
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英文标题：
《A model of returns for the post-credit-crunch reality: Hybrid Brownian
  motion with price feedback》
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作者：
William T. Shaw
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最新提交年份：
2009
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Applications        应用程序
分类描述：Biology, Education, Epidemiology, Engineering, Environmental Sciences, Medical, Physical Sciences, Quality Control, Social Sciences
生物学，教育学，流行病学，工程学，环境科学，医学，物理科学，质量控制，社会科学
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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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 "$25\sigma$" 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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PDF链接：
https://arxiv.org/pdf/0811.0182