We propose a continuous-time heterogeneous agent model consisting of fundamental, momentum, and contrarian traders to explain the significant time series momentum. We show that the performance of momentum strategy is determined by both time horizon and the market dominance of momentum traders. Specifically, when momentum traders are more active in the market, momentum strategies with short (long) time horizons stabilize (destabilize) the market, and meanwhile the market under-reacts (over-reacts) in short-run (long-run). This provides profit opportunity for time series momentum strategies with short horizons and reversal with long horizons. When momentum traders are less active in the market, they always lose. The results provide an insight into the profitability of time series momentum documented in recent empirical studies.

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.

By considering a financial market of fundamentalists and trend followers in which the price trend of the trend followers is formed as a weighted average of historical prices, we establish a continuous-time financial market model with time delay and examines the impact of time delay on market price dynamics. Conditions for the stability of the fundamental price in terms of agents' behavior parameters and time delay are obtained. In particular, it is found that an increase in time delay can not only destabilize the market price but also stabilize an otherwise unstable market price, leading to stability switching as delay increases. This interesting phenomena shed new light in understanding of mechanism on the market stability. When the fundamental price becomes unstable through Hopf bifurcations, sufficient conditions on the stability and global existence of the periodic solution are obtained.

We would like to thank Cars Hommes for a stimulating discussion in the early stages of this project. We are grateful to valuable comments from two anonymous referees. Dieci gratefully acknowledges a Visiting Professor Appointment at the Quantitative Finance Research Centre, UTS Business School, during which this work was finalised. Dieci also acknowledges support from MIUR under project PRIN 2009 "Local interactions and global dynamics in economics and finance: models and tools" and from EU COST within Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation". Financial support for Chiarella and He from the Australian Research Council (ARC) under Discovery Grant (DP110104487) is gratefully acknowledged.

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