AIMS Mathematics (Mar 2021)
Fat tails arise endogenously from supply/demand, with or without jump processes
Abstract
We show that the quotient of Levy processes of jump-diffusion type has a fat-tailed distribution. An application is to price theory in economics, with the result that fat tails arise endogenously from modeling of price change based on an excess demand analysis resulting in a quotient of arbitrarily correlated demand and supply whether or not jump discontinuities are present. The assumption is that supply and demand are described by drift terms, Brownian (i.e., Gaussian) and compound Poisson jump processes. If $P^{-1}dP/dt$ (the relative price change in an interval $dt$) is given by a suitable function of relative excess demand, $\left( \mathcal{D}-\mathcal{S}\right) /\mathcal{S}$ (where $\mathcal{D}$ and $\mathcal{S}$ are demand and supply), then the distribution has tail behavior $F\left( x\right) \sim x^{-\zeta}$ for a power $\zeta$ that depends on the function $G$ in $P^{-1}dP/dt=G\left( \mathcal{D}/\mathcal{S}\right) $. For $G\left( x\right) \sim\left\vert x\right\vert ^{1/q}$ one has $\zeta=q.$ The empirical data for assets typically yields a value, $\zeta\tilde{=}3,$ or $\zeta\in\left[ 3,5\right] $ for many financial markets. Many theoretical explanations have been offered for the disparity between the tail behavior of the standard asset price equation and empirical data. This issue never arises if one models price dynamics using basic economics methodology, i.e., generalized Walrasian adjustment, rather than the usual starting point for classical finance which assumes a normal distribution of price changes. The function $G$ is deterministic, and can be calibrated with a smaller data set. The results establish a simple link between the decay exponent of the density function and the price adjustment function, a feature that can improve methodology for risk assessment. The mathematical results can be applied to other problems involving the relative difference or quotient of Levy processes of jump-diffusion type.
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