Noisy Chaos in a Large System of Decision‐Makers with Heterogeneous Beliefs with Application to Index Option Prices

Abstract

Decision-makers (DMs) often exhibit preference reversal and phase transitions in their states of confidence. We show how those behaviors induce chaos in a large system of DMs with heterogeneous beliefs. We construct a behavioral matrix operator [from a DM's probability weighting function (PWF)] that characterizes a projective mapping of confidence in dual psychological spaces. Moreover, we prove that in a seemingly stable system of DMs, the empirical Lyapunov exponent process for the noisy orbit of DM deterministic PWFs admits tail event chaos controlled by risk attitude factors. We use reported results from rank-dependent Standard & Poor's 500 index option prices and Chicago Board Options Exchange Volatility Index to illustrate our theory. Data show phase transitions in the state of DM confidence over time as PWFs change from skew S-shape to inverted S-shape. Our model identifies an instability criterion function for the distribution of critical values of risk attitude factors over a convex set of probabilities bounded from above by the fixed-point probability for PWFs. In particular, we use the criterion function to estimate critical values for risk attitude factors in index option price data that predict market crash based on the orientation of the PWF implied by option data.