Making no-arbitrage discounting-invariant: a new FTAP beyond NFLVR and NUPBR

Abstract

In the simplest formulation, this paper addresses the following question: Given two positive asset prices on a right-open interval, how can one decide, in an economically natural manner, whether or not this is an arbitrage-free model?
In general multi-asset models of financial markets, the classic notions NFLVR and NUPBR depend crucially on how prices are discounted. To avoid such issues, we introduce a discounting-invariant absence-of-arbitrage concept. Like in earlier work, this rests on zero or some basic strategies being 'maximal'; the novelty is that maximality of a strategy is defined in terms of 'share' holdings instead of 'value'. This allows us to generalise both NFLVR, by dynamic share efficiency, and NUPBR, by dynamic share viability. These concepts are the same for discounted or undiscounted prices, and they can be used in open-ended models under minimal assumptions on asset prices. We establish corresponding versions of the FTAP, i.e., dual characterisations in terms of martingale properties. As one expects, “properly anticipated prices fluctuate randomly”, but with an 'endogenous' discounting process which must not be chosen a priori. The classic Black–Scholes model on [0,∞) is arbitrage-free in this sense if and only if its parameters satisfy m−r ∈ {0, σ²} or, equivalently, either bond-discounted or stock-discounted prices are martingales.

Keywords: absence of arbitrage, maximal strategies, semimartingales, discounting, NFLVR, NUPBR, FTAP, -martingale discounter, strongly share maximal, dynamic share viability, Black–Scholes model

JEL Classification: C00, G10