A Remark on Arbitrage and Martingale Measure

Abstract

In the option pricing theory, as was pointed out in Harrison-Kreps [6], the most important fact is that the absence of arbitrage follows from the existence of equivalent martingale measure for the price process of securities. There are several attempts to show the converse statement that the absence of arbitrage implies the existence of an equivalent martingale measure. In the discrete time case, the proof for the most general case has been given by Dalang-Morton-Willinger [2] and Schachermayer [9]. On the continuous time case Strieker [11] gave beautiful results, and they were extended by Delbaen [3], Schachermayer [10] and Delbaen-Schachermayer [4], [5]. In particular, the Mackey topology is cleverly used in Delbaen [3], and his result is quite satisfactory in the case where price processes are path-wise continuous. In this paper, we give some remarks on Orlicz spaces, Mackey topologies, and results by Ansel-Strieker [1] and Delbaen [3]. Then we give a certain necessary and sufficient conditions for the existence of an equivalent martingale measure.