Dynamics of moving average rules in a continuous-time financial market model

Abstract

Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used in discrete-time HAMs. The time delay represents a memory length of a moving average rule in discrete-time HAMs. Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and memory length are obtained. It is found that an increase in memory length not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize an otherwise unstable market price, leading to stability switching as the memory length increases. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized facts observed in financial markets.

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

“…This form is used in He and Zheng [6]. Concerning the specification of s(t), Qu and Wei [10] examine the case of n = 1 in which the trend includes one delay price with α 1 = 1…”

“…Two segments form an elliptical-shaped closed curve having the starting point S = (τ Since the delays have two constraints, τ i ≤ 5 for i = 1, 2 and τ 1 < τ 2 due to Assumptions 1 and 3, the feasible region should be above the diagonal line and subject to τ 2 ≤ 5. It is colored in yellow and further sub-divided into two sub-regions by the L 2 (1, 1) segment that intersects two lines, one is the diagonal at (τ e , τ e ) and the other is the horizontal line at τ 2 = 5 6 . It is clear from the yellow region that stability is preserved for τ 1 and τ 2 such as 0 < τ 1 < τ e and τ 1 < τ 2 < τ e .…”

“…Moreover, Chiarella [5], which is a development of Beja and Goldman [4], shows that the market price tends to a stable limit cycle under a nonlinear demand function of the risky asset when the equilibrium is unstable. More recently, He and Zheng [6] reconstruct the discretetime model of Chiarella et al [2] in a continuous-time framework in which the expected price is formed with a moving average of the past (delay) prices. 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.…”

An Analysis on the Fractional Asset Flow Differential Equations

“…Over the last three decades, the mathematical model has played an essential role in preparing planning to predict the financial behavior in the market in the future. Thus, a number of mathematical models have been used to predict financial behavior [1][2][3][4][5][6][7][8][9][10][11][12]. Moreover, in order to obtain better simulation of future phenomena, nonlinear continuous models have also developed.…”