No free lunch with vanishing risk


No free lunch with vanishing risk 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.

Mathematical representation

For a semimartingale S, let where a strategy is admissible if it is permitted by the market. Then define. S is said to satisfy no free lunch with vanishing risk if such that is the closure of C in the norm topology of.

Fundamental theorem of asset pricing

If is a semimartingale with values in then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure such that S is a sigma-martingale under.