
Investment Arbitrage
The following was implemented in Maple by Marcus Davidsson (2009)
davidsson_marcus@hotmail.com
and is based upon Chapter 7 in Bailey (2005) The Economic of Financial Markets

We assume that we have three assets which has the following initial prices

Initial Price

Asset A

3

Asset B

2

Asset C


We now assume that we have two different scenarios which leads to the following assets prices:

Asset A

Asset B

Asset C





Payoff Scenario One

10

8

9





Payoff Scenario Two

8

0

12





We now assume that we invest amount in asset A, amount in asset B and amount in asset C and that the amounts must sum to one

(1) 
Which means that

(2) 
Note that the individul weight does not have be positive. For example if we buy asset A then and if we short sell asset A then
We now note that the NoArbitrage Principle states that:
1) The Portfolio Cost must be zero
2) The Expected Portfolio Payoff at any time must be equal to zero (when some assets goes up some assets go down = zero payoff ).
The NoArbitrage Principle therefore gives us :
Portfolio Cost


Expected Portfolio Payoff Scenario One


Expected Portfolio Payoff Scenario Two


We can now simulataniously solve the two equations for the Expected Portfolio Payoff's in scenario one and two for and

(3) 
If we substitute in that into the Portfolio Cost equation and solve for we get:

(4) 
This means that the price of asset C must be equal to otherwise there would will exist an arbitrage opportunity.
Arbitrage1: Pc > 3
We now assume that the asset prices and payoff are the same as in the previous example and that which gives us:
We now assume that we buy two units of asset A, we short sell one unit of asset B and we short sell one unit of asset C so we get:
The Initial Portfolio Cost is therefore given by

(5) 
The Expected Portfolio Payoff for Scenario One is given by

(6) 
The Expected Portfolio Payoff for Scenario One is given by

(7) 
Which by definition is an arbitrage oportunity since the portfolio cost is zero and we are garanteed a posstive expected payoff iregardless
of which of the two scenarios are going to happen.
Arbitrage2: Pc< 3
We now assume that the asset prices and payoff are the same as in the previous example and that which gives us:
We now assume that we short sell one unit of asset A, we buy one unit of asset B and we buy one unit of asset C so we get:
The Initial Portfolio Cost is therefore given by

(8) 
The Expected Portfolio Payoff for Scenario One is given by

(9) 
The Expected Portfolio Payoff for Scenario One is given by

(10) 
Which again by definition is an arbitrage oportunity since the portfolio cost is zero and we are garanteed a posstive expected payoff iregardless
of which of the two scenarios are going to happen.