Asset Price Behavior in Complex Environments

This paper first surveys empirical patterns that appear in asset returns. The main patterns are the following. (i) Returns are hard to predict out of sample. The correlation dimension is rather large and is unstable; i.e., returns are generated by a “complex” process. (ii) Detrended volume of trading activity is very persistent; i.e., the autocorrelation function is positive and decreases slowly as lag length increases. (iii) Volatility of returns is also very persistent and the autocorrelation function of volatility is rather similar to that of detrended volume. (iv) Detrended volume and volatility are highly correlated contemporaneously. (v) The cross-autocorrelation function of detrended volume and volatility falls off rapidly in leads and lags with a large peak at 0. (vi) More sophisticated bootstrapping-type tests have found evidence of some conditional predictability of returns out of sample provided the conditioning set is chosen carefully. Secondly, this paper sketches development of adaptive evolutionary theoretic financial asset pricing models that nest the usual rational expectations type of models (e.g., a class of models that are versions of the Efficient Market Hypothesis). But rational expectations beliefs are costly and compete with other types of beliefs in generating net trading profits as the system moves forward in time. The paper briefly reviews some studies of equilibrium returns, volume of trading, and volatility of returns that is consistent with the stylized facts reviewed above. An “ensemble” approach is developed, rather in the spirit of statistical mechanics, which seems to be useful in developing analytical results for large evolutionary stochastic dynamical systems.