Estimating Stable Fixed Points and Langevin Potentials for Financial Dynamics

TL;DR

This paper extends the Geometric Brownian Motion model to include polynomial drift, demonstrating that a quadratic drift best fits financial data and reveals stable price points through potential well analysis.

Contribution

It generalizes the GBM to polynomial drift SDEs and shows quadratic drift as optimal, uncovering stable fixed points in financial dynamics.

Findings

Quadratic drift (q=2) is most frequently optimal for financial data.

Potential functions exhibit clear potential wells indicating stable prices.

Model selection favors polynomial drift of order two.

Abstract

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 generalises the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 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.