No free lunch with bounded and vanishing risk

Question

I am reading a book which states 'No free lunch with bounded risk as follows

where $\tilde{V}_t$ is the discounted value of the portfolio.Then it states the following theorem

EMM is the equivalent martingale measure.

But Wikipedia states the same theorem in the following way

Does this mean that the two conditions No free lunch with bounded risk and No free lunch with vanishing risk are equivalent. If yes how can I show it.

Answer 1

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 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 $\Rightarrow$ NFLVR.

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
• Their first example is a market with unique ELMM but which allows arbitrage strategies (NA). Thus, ELMM exists $\nRightarrow$ NA.
• Their second example is a market without arbitrage strategies and without an EMM. NFLBR is satisfied though. Thus, NFLBR $\nRightarrow$ EMM exists. A local martingale measures exists though.
• Back and Pliska (1991) also give an example of an arbitrage-free market without an EMM.