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 c\`adl\`ag 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 It\^o integration.