How to Build and Solve Continuous-Time Heterogeneous Agents Models in Asset Pricing? The Martingale Approach and the Finite Difference Method

Abstract

This paper shows step-by-step how to build and solve a preference heterogeneous agent model in asset pricing numerically when financial markets are complete. The proposed framework clarifies the modeling and solving process by an example and could be adapted to consider other heterogeneity types. In this framework, I also demonstrate that using the martingale technique to find the equilibrium, we need to deal with a partial differential equation (PDE), as in the dynamic programming technique. I then show how to implement the finite difference method with an upwind scheme to solve that PDE and hence obtain the equilibrium asset price and its volatility.

Keywords: heterogeneous agents, preferences, asset pricing, martingale, finite difference, continuous time