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 representationEdit

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 pricingEdit

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]


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