Fundamental Theorem of Discrete Asset Pricing

Abstract

Motivated by discrete price quotes in continuous time, we seek a fundamental theorem based on no arbitrage rather than a condition on limiting trading opportunities. We consider a vector of discounted prices taking values in a locally finite subset of $R^d$ in continuous time. For a locally bounded càdlàg price, we prove that no arbitrage, no free lunch with vanishing risk, no free lunch, and the existence of an equivalent local martingale measure are equivalent. An elementary finite-dimensional reduction of portfolios yields a common pathwise bound on admissible wealth. This bound reduces closedness to convex averaging in a weighted Hilbert space, after which the Kreps-Yan theorem applies. Trading gains are finite Stieltjes sums, and the proof uses no Itô integration.