No free lunch with vanishing risk

Summary

No free lunch with vanishing risk (NFLVR) is a concept used in mathematical finance as a strengthening of the no-arbitrage condition. In continuous time finance the existence of an equivalent martingale measure (EMM) is no more equivalent to the no-arbitrage-condition (unlike in discrete time finance), but is instead equivalent to the NFLVR-condition. This is known as the first fundamental theorem of asset pricing.

Informally speaking, a market allows for a free lunch with vanishing risk if there are admissible strategies, which can be chosen arbitrarily close to an arbitrage strategy, i.e., these strategies start with no wealth, end up with positive wealth with probability greater zero (free lunch) and the probability of ending up with negative wealth can be chosen arbitrarily small (vanishing risk).[1]

Mathematical definition edit

For a semimartingale  , let

  •   where a strategy is called admissible if it is self-financing and its value process   is bounded from below.
  •  .

  is said to satisfy the no free lunch with vanishing risk (NFLVR) condition if  , where   is the closure of C in the norm topology of  .[2]

A direct consequence of that definition is the following:

If a market does not satisfy NFLVR, then there exists   and sequences  ,   such that   and  . Moreover, it holds

  1.   (vanishing risk)
  2.   (free lunch)

In other words, this means: There exists a sequence of admissible strategies   starting with zero wealth, such that the negative part of their final values   converge uniformly to zero and the probabilities of the events  converge to a positive number.

Fundamental theorem of asset pricing edit

If   is a semimartingale with values in   then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure   such that S is a sigma-martingale under  .[3]

References edit

  1. ^ Delbaen, Schachermayer, Freddy, Walter (2008). The Mathematics of Arbitrage (corrected 2nd ed.). Berlin Heidelberg: Springer-Verlag. p. 78. ISBN 978-3-540-21992-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
  2. ^ Delbaen, Freddy; Schachermayer, Walter (2006). The mathematics of arbitrage. Vol. 13. Birkhäuser. ISBN 978-3-540-21992-7.
  3. ^ Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?" (PDF). Notices of the AMS. 51 (5): 526–528. Retrieved October 14, 2011.