Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing

Abstract

We consider an incomplete market model where asset prices are modelled by Ito processes, and derive the first fundamental theorem of asset pricing using standard stochastic calculus techniques. This contrasts with the sophisticated functional analytic theorems required in the comprehensive works of F. Delbaen and W. Schachermayer (1993) No Arbitrage and the Fundamental Theorem of Asset Pricing, pp. 37–38; Math. Finance 4 (1994), pp. 343–348; Math. Ann. 300 (1994), pp. 464–520; Ann. Appl. Probab. 5 (1995), pp. 926–645 and Proc. Sympos. Appl. Math. 57 (1999), pp. 49–58, and the comparative lack of transparency of the associated technical conditions. An additional benefit is that a clear relationship between no arbitrage and the existence of equivalent local martingale measures is also presented.