No arbitrage of the first kind and local martingale numéraires

Finance and Stochastics. 2016. Vol. 20. No. 4. P. 1097–1108.

Kabanov Y., Kardaras C., Song S.

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 (NA1NA1) 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 NA1NA1, a local martingale numéraire may fail to exist. In this work, we establish that under NA1NA1, a supermartingale numéraire under the original probability PP becomes a local martingale numéraire for equivalent probabilities arbitrarily close to PP in the total variation distance.

Keywords: arbitrage, viability, Fundamental theorem of asset pricing, Local martingale deflator, Numéraire, σ-martingale