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 (NA1$NA1$) 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 NA1$NA1$, a local martingale numéraire may fail to exist. In this work, we establish that under NA1$NA1$, a supermartingale numéraire under the original probability P$P$ becomes a local martingale numéraire for equivalent probabilities arbitrarily close to P$P$ in the total variation distance.

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