There are two different papers published by Freddy Delbaen and Walter Schachermayer in 1994.
A general version of the fundamental theorem of asset pricing
They prove a general version of the first fundamental theorem of asset pricing.
- Published in Mathematische Annalen
- They prove (Theorem 1.1 in the paper) that NFLVR is equivalent to the existence of at least one EMM (``First FTAP''). We need to differentiate three cases though, depending on the price process $S$ which we assume to a be a semimartingale (Wikipedia and your textbook refer to different versions of the same theorem.):
- If $S$ is bounded, then an equivalent measure exists under which $S$ is a martingale, see Delbaen and Schachermayer (1994, Theorem 1.1).
- If $S$ is locally bounded, then an equivalent measure exists under which $S$ is a local martingale, see Delbaen and Schachermayer (1994, Corollary 1.2).
- If $S$ is unbounded, then an equivalent measure exists under which $S$ is a sigma-martingale, see Delbaen and Schachermayer (1998, Theorem 1.1).
- In section 6, study the relationship between NFLBR, NFL and NFLVR. The gist is that NFLVR is a special case of NFLBR. Here's a quote from the paper (page 501):
The difference between (NFLVR) and (NFLBR) is now clear. In the no free lunch with vanishing risk property we deal with sequences such that the negative parts tend to 0 uniformly. In the no free lunch with bounded risk property we only require these negative parts to tend to 0 in probability and remain uniformly bounded!
- Thus, in general, NFLBR $\Leftarrow$$\Rightarrow$ NFLVR $\Leftrightarrow$ EMM exists.
The version of the FTAP from Delbaen and Schachermayer is amongst the most general versions of the first FTAP. The original ideas trace back to Ross (1978) and Harrison and Kreps (1979).
Arbitrage and free lunch with bounded risk for unbounded continuous processes
They give two examples of continuous, but unbounded semimartingales.
- Published in Mathematical Finance
- They give anTheir first example ofis a market without arbitrages that does not have an EMMwith unique ELMM but which allows arbitrage strategies (example 2 in the paperNA). Thus, ELMM exists $\nRightarrow$ NA.
- Their second example is a market without arbitrage strategies and without an EMM. NFLBR is satisfied though. Thus, this paper presents a counterexample to NFLBR $\Rightarrow$$\nRightarrow$ EMM exists. LocalA local martingale measures are more closely linked to NFLBR.
- NFLBR has also been used in other earlier papersexists though.
- Back and Pliska (1991) also give an example of an arbitrage-free market without an EMM.