An Advection-Difusion Model Incorporating Investor Inertia for the Dynamics of Financial Asset Prices

Abstract: Standard models of asset price dynamics, such as geometric Brownian motion (Osborne, 1959, Samuelson, 2016), do not formally incorporate investor inertia. This paper introduces a novel framework for modelling stock price dynamics that incorporates the concept of investor inertia, inspired by diffusion with retention models (Bevilacqua, 2011). The asset's log-price is modelled as a three-state discrete random walk, allowing for movements in any of three directions: up, down, or neutral. We demonstrate that this framework naturally leads to an advection-diffusion partial differential equation, in which the advection (drift) term arises directly from the asymmetry between buying, selling, and holding decisions. Remarkably, the model implies that log-prices follow a normal distribution a finding of great practical interest due to its analytical tractability. The applicability of the model is confirmed through simulation and an empirical application using Brazilian PETR4.SA data.