Estimating stable fixed points and Langevin potentials for financial dynamics

The geometric Brownian motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalizes the GBM to an SDE with polynomial drift of order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>q</a:mi></a:math> and shows via model selection that <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>q</b:mi><b:mo>=</b:mo><b:mn>2</b:mn></b:mrow></b:math> is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price. Published by the American Physical Society 2024