List of prime numbers: Difference between revisions

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The '''fundamental theorems of arbitrage/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. 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| 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]] (Ω, <math>\scriptstyle \mathcal{F}</math>, <math>\scriptstyle \mathcal{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 (finance)|bond]] ''B'' is [[Complete market|complete]] if and only if there exists a unique risk-neutral measure that is equivalent to ''P'' and has [[numéraire]] ''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]] 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=http://www.ams.org/notices/200405/what-is.pdf|format=pdf|accessdate=October 14, 2011}}</ref>
 
==See also==
*[[Arbitrage pricing theory]]
*[[Rational pricing]]
 
==References==
{{Reflist}}
*{{cite journal |last=Harrison |first=J. Michael |authorlink= |coauthors=Pliska, Stanley R. |year=1981 |month= |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 |url= |accessdate= |quote= }}
*{{cite journal |last=Delbaen |first=Freddy |authorlink= |coauthors=Schachermayer, Walter |year=1994 |month= |title=A General Version of the Fundamental Theorem of Asset Pricing |journal={{lang|de|Mathematische Annalen}} |volume=300 |issue=1 |pages=463–520 |doi=10.1007/BF01450498 |url= |accessdate= |quote= }}
 
==External links==
* http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf
 
{{DEFAULTSORT:Fundamental Theorem Of Arbitrage-Free Pricing}}
[[Category:Mathematical finance]]
[[Category:Fundamental theorems|Arbitrage-free pricing]]

Latest revision as of 01:51, 11 January 2015

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