# Finitely additive probabilities and the Fundamental Theorem of Asset Pricing

## TL;DR

This paper explores the connection between no arbitrage of the first kind in financial markets and finitely additive probabilities, providing new insights and proofs related to the Fundamental Theorem of Asset Pricing.

## Contribution

It establishes that absence of arbitrages of the first kind is equivalent to the existence of a specific finitely additive probability measure, offering a new perspective and proof of the FTAP.

## Key findings

- Existence of finitely additive probability under no arbitrage of the first kind
- New proof of the FTAP by Delbaen and Schachermayer
- Elementary treatment for continuous-path semimartingale models

## Abstract

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 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 FTAP of Delbaen and Schachermayer. Finally, an elementary and short treatment of the previous discussion is presented for the case of continuous-path semimartingale asset-price processes.