# Arbitrage Theorem

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, Optimizing 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 existence 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 existence of a non-linear return function, such as a stock option.

## Example

Suppose there are three horses in a race. You can place a bet for the winner at the following odds:

Outcome (horse that wins) Odds Probability implied by Odds Blackie 1:1 1/2 Brownie 2:1 1/3 Marko 3:1 1/4

Since the posted odds don't sum to 1, the arbitrage theorem implies that you should be able to place a series of wagers that guarantees a net winning in every outcome.

You place the following bets:

Outcome (horse that wins) Amount bet that horse loses Blackie -$10 Brownie -$7 Marko -$5

The second bet means that if Brownie wins, you pay $14 (i.e., 2:1 odds). But if he loses, you receive $7.

Outcome Amount received Blackie -$10+7+5 = $2 Brownie $10-$14+5 = $1 Marko $10+7-15 = $2

We win regardless of which outcome we chose. Hence, this is an arbitrage bet. The fact we can do this implies that there is no probability distribution on the outcomes of these horses that could result in the stated odds payouts.

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