Market Structure and Instability Artifacts in Heterogeneous Agent Models: Lessons from Implicit Discretizations of Stiff Equations

Abstract

We consider a standard heterogeneous agent model (HAM) that is widely used to analyze price developments in financial markets. The model is linear in log-prices and, in its basic setting, populated by fundamentalists and chartists. As the number of fundamentalists increases and exceeds a specific threshold, oscillations occur whose amplitude might even grow exponentially over time. From an economic perspective to adequately interpret such instability results it is indispensable to ensure that the characteristics and specific building blocks of the HAM are not at odds with the underlying structure of financial markets, in particular the specific trading rules. We expect that in markets with (almost) only fundamentalist traders prices might in the most extreme case oscillate, but never explode. In addition, if limit orders are available, prices should converge monotonically. Finally, if price bubbles occur in financial markets with fundamentalist traders, they should only result from the interactions between fundamentalists and the other traders, e.g., chartists, but not from fundamentalists’ decisions alone. From a mathematical perspective we show that the instability result common to the standard approach can be related to a “hidden” explicit discretization of a stiff ordinary differential equation contained in the model. Replacing this explicit discretization by an implicit one improves the model as it removes this artifact, bringing the model’s prediction in line with standard theory. The refined model still allows for price overshoots, bubbles, and crashes. However, in the implicit model these instabilities are caused by chartists and not by an unintended artifact.