The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete.
An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss.
[1] Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit.
[2]: 5 The first theorem is important in that it ensures a fundamental property of market models.
Though this property is common in models, it is not always considered desirable or realistic.
[2]: 30 In a discrete (i.e. finite state) market, the following hold:[2] When stock price returns follow a single Brownian motion, there is a unique risk neutral measure.
When the stock price process is assumed to follow a more general sigma-martingale or semimartingale, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk (NFLVR) must be used to describe these opportunities in an infinite dimensional setting.
[3] In continuous time, a version of the fundamental theorems of asset pricing reads:[4] Let