Finitely additive probabilities and the fundamental theorem of asset pricing

This work aims at a deeper understanding of the mathematical implications of the economically-sound condition of absence of arbitrages of the first kind in a financial market. In the spirit of the Fundamental Theorem of Asset Pricing (FTAP), it is shown here that the absence of arbitrages of the first kind in the market is equivalent to the existence of a finitely additive probability, weakly equivalent to the original and only locally countably additive, under which the discounted wealth processes become “local martingales”. The aforementioned result is then used to obtain an independent proof of the classical FTAP, as it appears in Delbaen and Schachermayer (Math. Ann. 300:463–520, 1994). Finally, an elementary and short treatment of the previous discussion is presented for the case of continuous-path semimartingale asset-price processes.