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 K = { ( H ⋅ S ) ∞ : H admissible , ( H ⋅ S ) ∞ = lim t → ∞ ( H ⋅ S ) t exists a.s. } where a strategy is admissible if it is permitted by the market. Then define C = { g ∈ L ∞ ( P ) : g ≤ f ∀ f ∈ K } . S is said to satisfy no free lunch with vanishing risk if C ¯ ∩ L + ∞ ( P ) = { 0 } such that C ¯ is the closure of C in the norm topology of L + ∞ ( P ) .
If S = ( S t ) t = 0 T is a semimartingale with values in R d then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure Q such that S is a sigma-martingale under Q .
