No free lunch with vanishing risk
Encyclopedia
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 portfolio
s which converge to an arbitrage strategy, we can approximate a self-financing portfolio (called the free lunch with vanishing risk).
S, let
where a strategy is admissible if it 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
.
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
.
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
argument. We have free lunch with vanishing risk if by utilizing a sequence of tame self-financing portfolio
Self-financing portfolio
Self-financing portfolio, an important concept in financial mathematics.A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one.- Mathematical definition :...
s which converge to an arbitrage strategy, we can approximate a self-financing portfolio (called the free lunch with vanishing risk).
Mathematical representation
For a semimartingaleSemimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....
S, let
where a strategy is admissible if it permitted by the marketFinancial market
In economics, a financial market is a mechanism that allows people and entities to buy and sell financial securities , commodities , and other fungible items of value at low transaction costs and at prices that reflect supply and demand.Both general markets and...
. Then define
. S is said to satisfy no free lunch with vanishing risk if such that is the closureClosure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of C in the norm topology of
.Fundamental theorem of asset pricing
If is a semimartingaleSemimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....
with values in
then S does not allow for a free lunch with vanishing risk if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there exists an equivalent martingale measure
such that S is a sigma-martingaleSigma-martingale
In the mathematical theory of probability, sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. It should be noted that not every local martingale is a sigma-martingale...
under
.