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).

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

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