Editing Fundamental theorem of asset pricing

{{short description|Necessary and sufficient conditions for a market to be arbitrage free and complete}}
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|arbitrage-free]], and for a market to be [[Complete market|complete]]. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss.<ref name="Varian">{{cite journal |title=The Arbitrage Principle in Financial Economics|first1=Hal R. |last1=Varian |author-link=Hal Varian|journal=Economic Perspectives |volume=1 |issue=2 |year=1987 |pages=55–72 |doi=10.1257/jep.1.2.55 |jstor=1942981| url=https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.1.2.55}}</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>{{rp|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 [[Replicating strategy|replicated]]. Though this property is common in models, it is not always considered desirable or realistic.<ref name=Pasc />{{rp|30}}

==Discrete markets==
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|arbitrage-free]] if, and only if, there exists at least one [[Risk-neutral measure|risk neutral probability measure]] that is [[Equivalence (measure theory)|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 market|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==

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>{{cite journal|title=What is... a Free Lunch?|first1=Freddy|last1=Delbaen|first2=Walter|last2=Schachermayer|journal=Notices of the AMS|volume=51|number=5|pages=526–528|url=https://www.ams.org/notices/200405/what-is.pdf|accessdate=October 14, 2011}}</ref>

In continuous time, a version of the fundamental theorems of asset pricing reads:<ref>{{Cite book |last=Björk |first=Tomas |title=Arbitrage Theory in Continuous Time |publisher=Oxford University Press |year=2004 |isbn=978-0-19-927126-9 |location=New York |pages=144ff |language=en}}</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 [[Risk-neutral measure|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 also==
*[[Arbitrage pricing theory]]
*[[Asset pricing]]
*{{section link|Financial economics #Arbitrage-free pricing and equilibrium}}
*[[Rational pricing]]

==References==
'''Sources'''
{{Reflist}}
'''Further reading'''
*{{cite journal |last=Harrison |first=J. Michael |author2=Pliska, Stanley R. |year=1981 |title=Martingales and Stochastic integrals in the theory of continuous trading |journal=Stochastic Processes and Their Applications |volume=11 |issue=3 |pages=215–260 |doi=10.1016/0304-4149(81)90026-0 |doi-access=free }}
*{{cite journal |last=Delbaen |first=Freddy |author2=Schachermayer, Walter |year=1994 |title=A General Version of the Fundamental Theorem of Asset Pricing |journal=Mathematische Annalen |volume=300 |issue=1 |pages=463–520 |doi=10.1007/BF01450498 }}

==External links==
* http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf

{{Portal bar|Business|Companies|Countries|Mathematics|Money}}

{{DEFAULTSORT:Fundamental Theorem Of Arbitrage-Free Pricing}}
[[Category:Financial economics]]
[[Category:Mathematical finance]]
[[Category:Corporate development]]