Fundamental theorem of asset pricing
Template:Short description 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.<ref name="Varian">Template:Cite journal</ref> Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit.<ref name=Pasc>Pascucci, Andrea (2011) PDE and Martingale Methods in Option Pricing. Berlin: Springer-Verlag</ref>Template:Rp 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.<ref name=Pasc />Template:Rp
Discrete marketsEdit
In a discrete (i.e. finite state) market, the following hold:<ref name=Pasc />
- The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space <math>(\Omega, \mathcal{F}, P)</math> 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 marketsEdit
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.<ref>Template:Cite journal</ref>
In continuous time, a version of the fundamental theorems of asset pricing reads:<ref>Template:Cite book</ref>
Let <math>S=(S_t)_{t\geq 0}</math> be a d-dimensional semimartingale market (a collection of stocks), <math>B</math> the risk-free bond and <math>(\Omega, \mathcal{F}, P)</math> the underlying probability space. Furthermore, we call a measure <math>Q</math> an equivalent local martingale measure if <math>Q \approx P</math> and if the processes <math>\left( \frac{S^i_t }{B_t} \right) _t</math> are local martingales under the measure <math>Q</math>.
- The First Fundamental Theorem of Asset Pricing: Assume <math>S</math> is locally bounded. Then the market <math>S</math> 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 <math>Q</math>. Then <math>S</math> is a complete market if and only if <math>Q</math> is the unique local martingale measure.
See alsoEdit
ReferencesEdit
Sources Template:Reflist Further reading