Let:

[math]\displaystyle{ i = 1..n }[/math] investments
[math]\displaystyle{ j = 1..m }[/math] outcomes
[math]\displaystyle{ v(i,j) }[/math] return from investment i when outcome j occurs
[math]\displaystyle{ p(j) }[/math] probability of outcome j
[math]\displaystyle{ x(i) }[/math] allocation (wager) on investment i
[math]\displaystyle{ r }[/math] risk free return

The Arbitrage Theorem

Exactly one of the following is true: Either:

(i) there exists a probability vector, [math]\displaystyle{ p(j) }[/math], for which for every i

[math]\displaystyle{ \sum_{j=1}^m p(j)v(i,j)=r }[/math]

(ii) there exists an allocation [math]\displaystyle{ x(i) }[/math] for which for every j

[math]\displaystyle{ \sum_{i=1}^m x(i)v(i,j) \gt r }[/math]

Description

This slide was shown during the User Group Webinar on 31-Mar-2011, [Parameters in a Complex Model to Match Historical Data].

Simply stated, the theorem states that either the expected value of all possible investments is the same and equal to the risk-free-rate of return, or it is possible to create an arbitrage portfolio that guarantees a return better than the risk-free rate of return for every possible outcome.

The theorem is a mathematical truth, which is proven by considering the relationship between the primary and dual of a linear program. Economists often apply the idea that no arbitrage should be possible in a perfect market when building theoretical models, and hence make use of this theorem. The existance of an arbitrage portfolio would imply a risk-free investment strategy that outperforms the risk-free rate of return. By assuming that such a portfolio does not exist, the theorem implies that all investments must have the same expected rate of return.

This theorem can be employed to obtain the well-known Black Scholes European option pricing model, for example. The Black-Scholes formula assumes that stocks follow a log-normal process and that arbitrage is not possible.

To apply the theorem, it must be possible to place a wager on each outcome independently, either positive or negative (i.e., long or short positions), without commissions overhead or price spreads. In a market scenario, that usually requires the existance of a non-linear return function, such as a stock option.