Abstract

We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific models— multiplicative, Heston (Cox-Ingersoll-Ross) and combined multiplicative-Heston—whose steady-state probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting “fat” tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function—in agreement with the general analysis— and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data.

Graphic abstract

中文翻译：

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历史市场数据的分布：松弛和相关性

摘要

我们研究一类随机变量的均值回复模型中的松弛和相关性。我们导出了相关函数的闭式表达式，并利用了随机项的一般形式。我们还讨论了三种特定模型（乘法，Heston（Cox-Ingersoll-Ross）和组合乘法-Heston）的相关函数和杠杆作用，它们的稳态概率密度函数分别为Gamma，Inverse Gamma和Beta Prime，后两个模型展示的是“肥”的尾巴。对于Heston模型，我们使用Fokker-Planck方程的特征值分析来推导相关函数（与一般分析一致），并确定一系列时间尺度，这些尺度在累积量趋于稳定的过程中可以观察到地弛豫。状态。

图形摘要