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).

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 $L_{+}^{\infty }(P)$.

Fundamental theorem of asset pricing

Main article: 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 $\mathbb {Q}$.