Weak and Strong No-Arbitrage Conditions for Continuous Financial Markets

Abstract

We propose a unified analysis of a whole spectrum of no-arbitrage conditions for financial market models based on continuous semimartingales. In particular, we focus on no-arbitrage conditions weaker than the classical notions of No Arbitrage and No Free Lunch with Vanishing Risk. We provide a complete characterisation of the considered no-arbitrage conditions, linking their validity to the characteristics of the discounted asset price process and to the existence and the properties of (weak) martingale deflators, and review classical as well as recent results.Keywords: arbitrage, benchmark approach, continuous semimartingale, martingale deflator, market price of risk, arbitrage of the first kind, free lunch with vanishing risk.MSC (2010): 60G44, 60H05, 91B70, 91G10.

Arbitrage-free pricing of derivatives in nonlinear market models

“…Nevertheless, we will argue that it can be effectively dealt with using some reasonably general definition of an arbitrage opportunity associated with trading. Let us stress that we only examine here a nonlinear extension of the classical concept of an arbitrage opportunity and hence the simplest definition of no-arbitrage, sometimes abbreviated as NA (see, for instance, part (iv) in Definition 2.2 in Fontana (2015)), as opposed to much more sophisticated concepts, such as: NFLVR (no free lunch with vanishing risk), NUPBR (no unbounded profit with bounded risk, which is also known as the no-arbitrage of the first kind, that is, NA1) or NIP (no increasing profit). The introduction of more sophisticated no-arbitrage conditions is motivated by the desire to establish a suitable version of the fundamental theorem of asset pricing (FTAP), which shows the equivalence between a particular form of no-arbitrage and the existence of some kind of a "martingale measure" for the discounted prices of primary assets.…”

“…One first checks whether a market model with predetermined trading rules and primary traded assets is arbitrage-free, where the definition of an arbitrage opportunity is a mathematical formalization of the real-world concept of a risk-free profitable trading opportunity. In fact, depending on the framework at hand, several alternative definitions of "no-arbitrage" were studied (for an overview, see Fontana (2015)). …”

Deterministic Criteria for the Absence and Existence of Arbitrage in Multi-Dimensional Diffusion Markets

“…The existence of a SLMD is equivalent to (NUPBR), see Choulli & Stricker (1996) and, in complete settings, the existence of a SMD is equivalent to (NRA), see Fernholz & Karatzas (2010). For a profound discussion of weak notions of no arbitrage we refer to the article of Fontana (2015).…”

No arbitrage in continuous financial markets

“…As observed by Mijatović and Urusov [46], for one dimensional diffusion models the (NRA) condition is not necessarily implied by (NFLVR). For further notions of no arbitrage we refer to the article of Fontana [26].…”