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. Completeness is a common property of market models (for instance the Black–Scholes model). A complete market is one in which every contingent claim can be replicated. 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]}

**The First Fundamental Theorem of Asset Pricing**: A discrete market on a discrete probability space is arbitrage-free if, and only if, there exists at least one risk neutral probability measure that is equivalent to the original probability measure,*P*.**The Second Fundamental Theorem of Asset Pricing**: An arbitrage-free market (S,B) consisting of a collection of stocks*S*and a risk-free bond*B*is complete if and only if there exists a unique risk-neutral measure that is equivalent to*P*and has numeraire*B*.

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 be a d-dimensional semimartingale market (a collection of stocks), the risk-free bond and the underlying probability space. Furthermore, we call a measure an equivalent local martingale measure if and if the processes are local martingales under the measure .

**The First Fundamental Theorem of Asset Pricing**: Assume is locally bounded. Then the market satisfies NFLVR if and only if there exists an equivalent local martingale measure.**The Second Fundamental Theorem of Asset Pricing**: Assume that there exists an equivalent local martingale measure . Then is a complete market if and only if is the unique local martingale measure.