Fundamental theorem of asset pricing
Necessary and sufficient conditions for a market to be arbitrage free and complete From Wikipedia, the free encyclopedia
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
Discrete markets
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.
In more general markets
Summarize
Perspective
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.
See also
References
External links
Wikiwand - on
Seamless Wikipedia browsing. On steroids.