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'''No free lunch with vanishing risk''' ('''NFLVR''') is a no-[[arbitrage]] argument.  We have ''free lunch with vanishing risk'' if by utilizing a sequence of tame [[self-financing portfolio]]s which converge to an arbitrage strategy, we can approximate a self-financing portfolio (called the ''free lunch with vanishing risk'').<ref>{{cite journal|first=Michael|last=Dothan|title=Efficiency and Arbitrage in Financial Markets|journal=International Research Journal of Finance and Economics|issue=19|year=2008|url=http://www.eurojournals.com/irjfe_19_08.pdf|format=pdf|accessdate=February 5, 2011}}</ref>

==Mathematical representation==
For a [[semimartingale]] ''S'', let <math>K = \{(H \cdot S)_{\infty}: H \text{ admissible}, (H \cdot S)_{\infty} = \lim_{t \to \infty} (H \cdot S)_t \text{ exists a.s.}\}</math> where a strategy is [[admissible trading strategy|admissible]] if it is permitted by the [[Financial market|market]].  Then define <math>C = \{g \in L^{\infty}(P): g \leq f \; \forall f \in K\}</math>.  ''S'' is said to satisfy ''no free lunch with vanishing risk'' if <math>\bar{C} \cap L^{\infty}_+(P) = \{0\}</math> such that <math>\bar{C}</math> is the [[Closure (topology)|closure]] of ''C'' in the [[norm topology]] of <math>L^{\infty}_+(P)</math>.<ref>{{cite book|first1=Freddy|last1=Delbaen|first2=Walter|last2=Schachermayer|title=The mathematics of arbitrage|volume=13|publisher=Birkhäuser|year=2006|isbn=978-3-540-21992-7}}</ref>

==Fundamental theorem of asset pricing==
{{main|fundamental theorem of asset pricing}}
If <math>S = (S_t)_{t=0}^T</math> is a [[semimartingale]] with values in <math>\mathbb{R}^d</math> then ''S'' does not allow for a free lunch with vanishing risk [[if and only if]] there exists an [[equivalent martingale measure]] <math>\mathbb{Q}</math> such that ''S'' is a [[sigma-martingale]] under <math>\mathbb{Q}</math>.<ref>{{cite journal|title=What is... a Free Lunch?|first1=Freddy|last1=Delbaen|first2=Walter|last2=Schachermayer|journal=Notices of the AMS|volume=51|number=5|pages=526–528|url=http://www.ams.org/notices/200405/what-is.pdf|format=pdf|accessdate=October 14, 2011}}</ref>

==References==
{{Reflist}}


[[Category:Financial markets]]
[[Category:Mathematical finance]]

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