No free lunch with vanishing risk (NFLVR) is a no-

arbitrage In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between the ...

argument. We have ''free lunch with vanishing risk'' if by utilizing a sequence of time

self-financing portfolio In financial mathematics, a self-financing portfolio is a portfolio having the feature that, 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

semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...

''S'', let

K = \

where a strategy is admissible if it is permitted by the

market Market is a term used to describe concepts such as: *Market (economics), system in which parties engage in transactions according to supply and demand *Market economy *Marketplace, a physical marketplace or public market Geography *Märket, an ...

. Then define

C = \

. ''S'' is said to satisfy ''no free lunch with vanishing risk'' if

\bar \cap L^_+(P) = \

such that

\bar

is the closure of ''C'' in the norm topology of

L^_+(P)

Fundamental theorem of asset pricing

S = (S_t)_^T

is a

with values in

\mathbb^d

then ''S'' does not allow for a free lunch with vanishing risk

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

there exists an equivalent martingale measure

\mathbb

such that ''S'' is a

sigma-martingale In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. In financial mathematics, sigma-martingale ...

under

\mathbb

References

Arbitrage Financial markets Mathematical finance {{Finance stub