A note on the condition of no unbounded profit with bounded risk

As a corollary to Delbaen and Schachermayer’s fundamental theorem of asset pricing (Delbaen in Math. Ann. 300:463–520, <CitationRef CitationID="CR5">1994</CitationRef>; Stoch. Stoch. Rep. 53:213–226, <CitationRef CitationID="CR6">1995</CitationRef>; Math. Ann. 312:215–250, <CitationRef CitationID="CR7">1998</CitationRef>), we prove, in a general finite-dimensional semimartingale setting, that the no unbounded profit with bounded risk (NUPBR) condition is equivalent to the existence of a strict sigma-martingale density. This generalizes the continuous-path result of Choulli and Stricker (Séminaire de Probabilités XXX, pp. 12–23, <CitationRef CitationID="CR2">1996</CitationRef>) to the càdlàg case and extends the recent one-dimensional result of Kardaras (Finance and Stochastics 16:651–667, <CitationRef CitationID="CR20">2012</CitationRef>) to the multidimensional case. It also refines partially the second main result of Karatzas and Kardaras (Finance Stoch. 11:447–493, <CitationRef CitationID="CR18">2007</CitationRef>) concerning the existence of an equivalent supermartingale deflator. The proof uses the technique of numéraire change. Copyright Springer-Verlag Berlin Heidelberg 2014