A Damped Diffusion Framework for Financial Modeling and Closed-Form Maximum Likelihood Estimation

Abstract

Asset price bubbles can arise unintentionally when one uses continuous-time diffusion processes to model financial quantities. We propose a flexible damped diffusion framework that is able to break many types of bubbles and preserve the martingale pricing approach. Damping can be done on either the diffusion or drift function. Oftentimes, certain solutions to the valuation PDE can be ruled out by requiring the solution to be a limit of martingale prices for damped diffusion models. Monte Carlo study shows that with finite time-series length, maximum likelihood estimation often fails to detect the damped diffusion function while fabricates nonlinear drift function. An alternative method based on Aı¨t-Sahalia’s specification test on parametric models is proposed.

Keywords: Damped diffusion, Asset Price Bubbles, Martingale Pricing, Maximum Likelihood Estimation

JEL Classification: C60, G12, G13