Martingale Restrictions on Equilibrium Prices of Arrow-Debreu Securities Under Rational Expectations and Consistent Beliefs

Consider the Rational Expectations price history of an Arrow-Debreu security that matures in the money: p(1), p(2), ..., p(T). Past information can be used to predict the return (p(t+1)-p(t))/p(t). Now consider a simple alternative performance measure: (p(t+1)-p(t))/p(t+1). It differs from the return only in that the future price is used as basis. This variable cannot be forecasted from past information. The result obtains even if investors' beliefs are biased, i.e., prices are not set in a Rational Expectations Equilibrium (REE). It depends only on investors' using the rules of conditional probability to process information. More precisely, the result continues to hold in the Bayesian Equilibrium with Consistent Beliefs (CBE) introduced by Harsanyi [1967]. Many related results are proved in this paper