# No free lunch with vanishing risk

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 time self-financing portfolios which converge to an arbitrage strategy, we can approximate a self-financing portfolio (called the free lunch with vanishing risk).[1]

## Mathematical representation

For a semimartingale S, let ${\displaystyle K=\{(H\cdot S)_{\infty }:H{\text{ admissible}},(H\cdot S)_{\infty }=\lim _{t\to \infty }(H\cdot S)_{t}{\text{ exists a.s.}}\}}$ where a strategy is admissible if it is permitted by the market. Then define ${\displaystyle C=\{g\in L^{\infty }(P):g\leq f\;\forall f\in K\}}$. S is said to satisfy no free lunch with vanishing risk if ${\displaystyle {\bar {C}}\cap L_{+}^{\infty }(P)=\{0\}}$ such that ${\displaystyle {\bar {C}}}$ is the closure of C in the norm topology of ${\displaystyle L_{+}^{\infty }(P)}$.[2]

## Fundamental theorem of asset pricing

If ${\displaystyle S=(S_{t})_{t=0}^{T}}$ is a semimartingale with values in ${\displaystyle \mathbb {R} ^{d}}$ then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure ${\displaystyle \mathbb {Q} }$ such that S is a sigma-martingale under ${\displaystyle \mathbb {Q} }$.[3]

## References

1. ^ Dothan, Michael (2008). "Efficiency and Arbitrage in Financial Markets" (pdf). International Research Journal of Finance and Economics (19). Retrieved February 5, 2011.
2. ^ Delbaen, Freddy; Schachermayer, Walter (2006). The mathematics of arbitrage. 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.