Clarifying Some Basic Concepts and Results in Arbitrage Pricing Theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan(1998) introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we will establish the Kramkov’s optional decomposition theorem in the setting of equivalent martingale measures and using this theorem to clarify some basic concepts and results in arbitrage pricing theory: superhedging, fair price, replicatable contingent claim, complete markets.