How Bounded-Risk and Vanishing-Risk No-Arbitrage Conditions Differ
Summary
The document clarifies how no free lunch with bounded risk (NFLBR) relates to no free lunch with vanishing risk (NFLVR), and why textbook statements of the fundamental theorem of asset pricing may differ. It distinguishes two Delbaen–Schachermayer papers and explains that the theorem’s conclusion depends on assumptions about the price process. For bounded prices, the equivalent measure makes prices martingales; for locally bounded prices, it makes them local martingales; for unbounded prices, the cited result uses a sigma-martingale.
The central distinction concerns sequences of portfolio outcomes: NFLVR requires negative parts to converge uniformly to zero, while NFLBR allows convergence in probability, provided those negative parts remain uniformly bounded. The answer states that NFLBR implies NFLVR in general, so the conditions are not equivalent in general. It also describes examples showing that, for unbounded continuous processes, an equivalent local martingale measure need not rule out arbitrage, and NFLBR need not ensure an equivalent martingale measure. These are theorem-level results whose applicability depends on the model’s assumptions; the document does not develop the proofs.
Key ideas
- NFLBR and NFLVR impose different conditions on the negative parts of portfolio outcomes.
- The document states that NFLBR implies NFLVR, but they are not generally equivalent.
- The fundamental theorem’s martingale conclusion depends on whether prices are bounded, locally bounded, or unbounded.
- For unbounded processes, existence of a local martingale measure does not by itself rule out arbitrage.
- The cited examples and theorem statements rely on their specified market-model assumptions.
Tags
Full text
# No free lunch with bounded and vanishing risk
# No free lunch with bounded and vanishing risk
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 by Kevin (score 3, accepted)
https://quant.stackexchange.com/a/66545
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.