Heterogeneous beliefs and adaptive behaviour in a continuous-time asset price model

Abstract

This paper extends the analysis of the seminal work of Brock and Hommes (1997, 1998) on heterogeneous beliefs and rational routes to randomness in discrete-time models to a continuous-time model of asset pricing. The resulting model characterized mathematically by a system of stochastic delay differential equations provides a unified approach to deal with adaptive behaviour of heterogeneous agents and market stability impact of lagged price used by chartists to form their expectations. For the underlying deterministic model, we show not only that the result of Brock and Hommes on rational routes to market instability in discrete-time holds in continuous-time but also a double edged effect of an increase in lagged price used by the chartists on market stability. For the stochastic model, we demonstrate that the interaction and boundedly rational behaviour of heterogeneous agents can generate various market phenomena such as bubbles and crashes and replicate stylized facts including volatility clustering, and long range dependence in volatility.

A continuous time Cournot duopoly with delays

“…This last consideration is closely related to the (still open) debate about whether it is better to build on models in discrete time or continuous time to describe and explain some economic phenomena, especially in financial markets (He and Zheng, 2010;He and Li, 2012). Indeed, this debate is not only philosophical but there may produce relevant differences in both the mathematical properties (especially under bounded rationality of agents) and final outcomes of models.…”

A Heterogeneous Agent Model of Asset Price with Three Time Delays

“…Their main result concerns a double edge effect on the stability caused by the length of the memory or delay: an increase in delay can destabilize the market price and also stabilize it. In their model as well as in subsequent studies on continuous-time HAMs such as He and Li [7] and Xu et al [8], this interesting phenomenon is obtained under the assumption that infinitely many past price data are available at no charge of cost. Needless to remember the well-known line in economics, "there is no such thing as a free lunch, " it is to go too far to get necessary information for nothing.…”

“…So the black segment depicted in the yellow region is defined for ω ∈ [ω m , ω 2 ]. 7 It is possible to obtain this value by following the same procedure with which we obtained the value of τ m 1 just above. So the stationary point p * = f is stable and the corresponding part of the bifurcation diagram is a horizontal line at p(t) = f .…”

“…When the varying pair (τ 1 , τ 2 ) with τ 2 > τ e crosses the L 2 (1, 1) segment, we have the stability switching as the real part of at least one eigenvalue turns to be positive and then the stationary point loses stability. To see this, we perform a simulation by increasing the value of τ 1 along the dotted horizontal line at τ 2 = 4 that crosses the L 2 (1, 1) curve at point (τ c 1 , 4) with τ c 1 ≃ 1.640 7 . The numerical results are summarized in Figure 4B in which a bifurcation diagram with respect to τ 1 is depicted.…”

News-Driven Expectations and Volatility Clustering

“…This is the case in this paper's model as well, as emphasized below (Figure 7). The reviewer nonetheless suggests that the white-noise assumption, along with traders' heterogeneity, may be responsible for clustered volatility in the agent-based model by He and Li (2012).…”