No arbitrage of the first kind and local martingale numéraires

Abstract

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind ( $\mbox{NA}_{1}$ ) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under $\mbox{NA}_{1}$ , a local martingale numéraire may fail to exist. In this work, we establish that under $\mbox{NA}_{1}$ , a supermartingale numéraire under the original probability $P$ becomes a local martingale numéraire for equivalent probabilities arbitrarily close to $P$ in the total variation distance.