No arbitrage of the first kind and local martingale numéraires

Yuri Kabanov (), Constantinos Kardaras () and Shiqi Song ()
Additional contact information
Yuri Kabanov: Université de Franche-Comté
Constantinos Kardaras: London School of Economics
Shiqi Song: Université d’Evry Val d’Essonne
Authors registered in the RePEc Author Service: Юрий Михайлович Кабанов

Finance and Stochastics, 2016, vol. 20, issue 4, No 9, 1097-1108

Abstract: 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 ( NA 1 $\mbox{NA}_{1}$ ) 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 NA 1 $\mbox{NA}_{1}$ , a local martingale numéraire may fail to exist. In this work, we establish that under NA 1 $\mbox{NA}_{1}$ , 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; Numéraire; Local martingale deflator; σ $\sigma$ -martingale; 91G10; 60G44 (search for similar items in EconPapers)
JEL-codes: C60 G13 (search for similar items in EconPapers)
Date: 2016
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (19)

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DOI: 10.1007/s00780-016-0310-6

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