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

Abstract

In general multi-asset models of financial markets, the classic no-arbitrage concepts NFLVR and NUPBR have a serious shortcoming — they depend crucially on the way prices are discounted. To avoid this unnatural economic behaviour, we introduce a new idea for defining “absence of arbitrage”. It rests on the new notion of strongly index weight maximal strategies, which allows us to generalise both NFLVR (by dynamic index weight efficiency) and NUPBR (by dynamic index weight viability). These new no-arbitrage concepts do not change when we look at discounted or undiscounted prices, and they can be used in open-ended models under very weak assumptions on asset prices. We establish corresponding versions of the FTAP, i.e., dual characterisations of our concepts in terms of martingale properties. A key new feature is that as one expects, “properly anticipated prices fluctuate randomly”, but with an endogenous discounting process which is not a priori chosen exogenously. We also illustrate our results by a wide range of examples. In particular, we show that the classic Black–Scholes model on [0,1) is arbitrage-free in our sense if and only if its parameters satisfy m−r e {0, σ²} or, equivalently, either bond-discounted or stock-discounted prices are martingales.