Pricing by Arbitrage Under Arbitrary Information

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities' normalized price histories are priced by arbitrage if and only if all risk-adjusted probabilities agree upon the law of primitive securities' normalized prices. We show how existing diffusion-based results can be preserved, and resolve an issue relating to Merton's (1973) stochastic interest rate model.