Complexity and the Character of Stock Returns: Empirical Evidence and a Model of Asset Prices Based on Complex Investor Learning
Empirical evidence on the distributional characteristics of common stock returns indicates: (1) A power-law tail index close to three describes the behavior of the positive tail of the survivor function of returns (pr(r > x) ~ x<sup> -\alpha </sup>), a reflection of fat tails; (2) general linear and nonlinear dependencies exist in the time series of returns; (3) the time-series return process is characterized by short-run dependence (short memory) in both returns as well as their volatility, the latter usually characterized in the form of autoregressive conditional heteroskedasticity; and (4) the time-series return process probably does not exhibit long memory, but the squared returns process does exhibit long memory. We propose a model of complex, self-referential learning and reasoning amongst economic agents that jointly produces security returns consistent with these general observed facts and which are supported here by empirical results presented for a benchmark sample of 50 stocks traded on the New York Stock Exchange. The market we postulate is populated by traders who reason inductively while compressing information into a few fuzzy notions that they can in turn process and analyze with fuzzy logic. We analyze the implications of such behavior for the returns on risky securities within the context of an artificial stock market model. Dynamic simulation experiments of the market are conducted, from which market-clearing prices emerge, allowing us to then compute realized returns. We test the effects of varying values of the parameters of the model on the character of the simulated returns. The results indicate that the model proposed in this paper can jointly account for the presence of a power-law characterization of the positive tail of the survivor function of returns with exponent on the order of three, for autoregressive conditional heteroskedasticity, for long memory in volatility, and for general nonlinear dependencies in returns.
