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

Fundamental theorem of asset pricing

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