Discounting-invariant absence of arbitrage at the level of NUPBR

Abstract

We consider a financial market modelled by a general

-valued semimartingale S on 

left bracket 0 comma normal infinity right parenthesis [ 0 , ∞ ) $[0, \infty)$

, without assuming the existence of a risk-free asset (savings account). We introduce a new absence-of-arbitrage concept, called

ModifyingBelow NUPBR With bar NUPBR _ ${\underline{\textrm{NUPBR}}}$

, which is a natural extension of the classic no unbounded profit with bounded risk (NUPBR) condition and which is in addition discounting-invariant for a large (but not maximal) class of discounters. We establish a dual characterisation of

ModifyingBelow NUPBR With bar NUPBR _ ${\underline{\textrm{NUPBR}}}$

in terms of martingale properties for S , and we prove the existence of an ‘optimal’ self-financing value process which generalises the numeraire portfolio from the classic mathematical finance literature. We further introduce a notion of maximality for value processes and provide for that a dual characterisation which extends existing concepts to settings where value processes may fail to converge as

t right arrow normal infinity t → ∞ $t\to\infty$

.