A continuous heterogeneous-agent model for the co-evolution of asset price and wealth distribution in financial market

The heterogeneous agent model (HAM) is a powerful tool to study the stock price dynamics and stylized anomalies in financial markets, such as the volatility clustering and long-range dependence of stock returns. The existing HAMs often require the perfect arbitrage condition and relatively strong behavioral assumptions on agents’ expectation formation, which are not sufficient to capture the complexity in real finance market. To fill the theoretical gap, we propose a novel continuous version of HAM that relies on weaker assumptions, in which heterogeneous agents are identified continuously by their wealth and the ex-post price expectations no matter how they are formed. Given the non-parametric distribution on the continuous agent space and the bounded arbitrage condition, we establish the joint evolution equations for stock price and wealth distribution. Within the family of mixed-exponential distributions, we derive an explicit expression for the equations and prove that the evolution equation of wealth distribution has the fascinating splitting-mixing property. An endogenous generating mechanism for agent types arises naturally from the splitting-mixing property, which links the continuous HAM to the classical HAMs with finite agent types. By numerical simulation, we show that the wealth dynamics tends to positively associate the weight of each generated agent type with their averaged investment performance, while the price dynamics experiences the period-doubling bifurcation and chaotic bifurcation. By introducing random shocks into the continuous HAM, we fit the resulting return series via a GARCH model. Based on the significance of the fitted GARCH parameters, a wide range of model parameters are identified by which the continuous HAM can simultaneously generate volatility clustering, long-range dependence and other stylized anomalies in financial markets.