The Boltzmann Equation in Finance

Abstract

This article seeks to bypass the reliance on the Kolmogorov Partial Differential Equations (PDEs) typically rooted in Markov stochastic processes by proposing a more flexible formula to represent different functions of random variables. This novel framework will result in the formulation of integro-differential and integro-difference equations, also referred to as Boltzmann equations, in the space of probability distributions. This approach offers greater flexibility as it governs several kinds of stochastic processes, including cases of both diffusion and concentration. Additionally, this framework allows for the derivation of the probability distribution of the price of a European contingent claim at maturity.

Keywords: Boltzmann Equation, Dirac Delta, Function of Random Variables, Conservation, Heaviside Function, Integral Operator, Integro-differential Equations, Call Option