正态随机变量的商及其在资产价格厚尾中的应用
The Quotient of Normal Random Variables And Application to Asset Price Fat Tails
摘要：
检验了正态分布随机变量的商，证明了其具有幂律衰减，密度为$f\left( x\right) \simeq f_{0}x^{-2}$，系数取决于分子和分母的均值和方差及其相关性.我们还获得了条件概率密度的四个象限的分子和分母的符号为任意相关$ρ\in\lbrack-1,1).$对于$ρ=-1$，我们得到了一个特别简单的封闭形式的解决方案，所有$x\in$ $\mathbb{R}$。结果被应用到经济学和金融学的一个基本问题，即相对价格变化的密度。古典金融学规定了相对价格变化的正态分布，尽管实证研究表明末端存在幂律。通过考虑基本价格变化模型中的供给和需求，我们证明了相对价格变化具有密度，该密度按照$x^{-2}$幂律衰减。确定了各种参数限值。
英文摘要：
The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density $f\left( x\right) \simeq f_{0}x^{-2}$, with the coefficient depending on the means and variances of the numerator and denominator and their correlation. We also obtain the conditional probability densities for each of the four quadrants given by the signs of the numerator and denominator for arbitrary correlation $ρ\in\lbrack-1,1).$ For $ρ=-1$ we obtain a particularly simple closed form solution for all $x\in$ $\mathbb{R}$. The results are applied to a basic issue in economics and finance, namely the density of relative price changes. Classical finance stipulates a normal distribution of relative price changes, though empirical studies suggest a power law at the tail end. By considering the supply and demand in a basic price change model, we prove that the relative price change has density that decays with an $x^{-2}$ power law. Various parameter limits are established.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
