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 $K = \{(H \cdot S)_{\infty}: H \text{ admissible}, (H \cdot S)_{\infty} = \lim_{t \to \infty} (H \cdot S)_t \text{ exists a.s.}\}$ where a strategy is admissible if it is permitted by the market. Then define $C=\{g\in L^{\infty }(P):\exists f\in K,~g\leq f~a.s.\}$ . 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

↑ Dothan, Michael (2008). "Efficiency and Arbitrage in Financial Markets" (pdf). International Research Journal of Finance and Economics (19). Retrieved February 5, 2011.
↑ Delbaen, Freddy; Schachermayer, Walter (2006). The mathematics of arbitrage. 13. Birkhäuser. ISBN 978-3-540-21992-7.
↑ Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?" (pdf). Notices of the AMS. 51 (5): 526–528. Retrieved October 14, 2011.

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[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.