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5.4 Linear Programming and Arbitrage 

Consider a portfolio Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi( Such a portfolio is self financed. The absolute percentage invested in the short position and in the long position is equal. Hence, the amount invested in long positions equals that in short positions and thus such a portfolio does not require out of pocket financing.    

The arbitrage in this model is in terms of risk and (expected) rate of return.  


The premise is that if there exists securities such that their expected rate of return does not fully compensate the holder for the risk borne or over compensate for it, a riskless self financed portfolio with positive (expected) rate of return exists.  The existence of such a portfolio means that the investor in this market can have an investment instrument that requires no money while producing positive expected rate of return. Such an opportunity will be exploited by investors generating a price pressure that will increase the price of the securities that are in a long position in this portfolio and decreases the prices of the securities that are held in a short position.  Consequently prices will adjust such that the opportunity to obtain positive (expected) rate of return with zero money and no risk-arbitrage opportunities will disappear. Hence the assumption is that such a portfolio does not exist in the market. The absence of such a portfolio implies the required result.  


The identification of such a portfolio can be formulated as a linear programming maximization problem. The objective function is the (expected) rate of return from the portfolio and the constraints ensure that the portfolio is riskless and self financed.  Since the optimization problem requires the portfolio to be riskless its expected rate of return is its rate of return, and thus we refer to it as (expected) rate of return. Hence the optimization problem is given by 

s.t.  Typesetting:-mrow(Typesetting:-mi( for  j=1,...,s

In this problem we are looking for a portfolio with a risk profile of (0,...,0).  

Each of the Typesetting:-mrow(Typesetting:-mi( constraints of the type  Typesetting:-mrow(Typesetting:-mi( ensures that the portfolio is riskless.  

The rate of return of the portfolio is  



which as we explained before can be written as 




where Typesetting:-mrow(Typesetting:-mi(and Typesetting:-mrow(Typesetting:-mi(However, since each of the Typesetting:-mrow(Typesetting:-mi( s are multiplied by Typesetting:-mrow(Typesetting:-mi( which is 0, this is a non-risky portfolio (to the extent that Typesetting:-mrow(Typesetting:-mi( is indeed to be zero). Thus its rate of return, which equals its expected rate of return since it is a riskless portfolio, is Typesetting:-mrow(Typesetting:-mi( Moreover, the constraint Typesetting:-mrow(Typesetting:-mi( ensures that the cost of this portfolio is zero.  If an optimal solution to this optimization problem with Typesetting:-mrow(Typesetting:-mi( not being zero exists, there exists in this market a self financed portfolio producing a positive risk free rate of return.  


Let us take a second look at this optimization problem. It is obviously a linear programming problem since all the involved functions are linear. Furthermore its optimal value can either be zero or it will be unbounded. If a feasible solution for which Typesetting:-mrow(Typesetting:-mi( is positive (negative) exists scaling this solution will increase Typesetting:-mrow(Typesetting:-mi( without bound. If the problem is unbounded it means that a portfolio whose (expected) return is infinite with no cost and bears no risk, exists in the market. Equilibrium cannot prevail in such a market and hence we assume that such a portfolio does not exist and therefore assume that the optimal value of this problem must be zero. This is the assumption made by the APT, namely the absence of arbitrage opportunities. 


The feasible set of the optimization problem consists of all the vectors that are perpendicular to the Typesetting:-mrow(Typesetting:-mi( vectors; the Typesetting:-mrow(Typesetting:-mi( dimensional vector  


and the Typesetting:-mrow(Typesetting:-mi(vectors:    




As we just explained the optimal value of the objective function, if there are no arbitrage opportunities, must be zero.  However this will be the case if and only if vector Typesetting:-mrow(Typesetting:-mi( must be in the subspace spanned by the Typesetting:-mrow(Typesetting:-mi(vectors above. That is, there must exist an Typesetting:-mrow(Typesetting:-mi(dimensional vector Typesetting:-mrow(Typesetting:-mi(such that 



One of the end of chapter questions asks the reader to prove the above statement. It is also proven at the end of this section with the aid of the Lagrangian which shows that the Typesetting:-mrow(Typesetting:-mi(are the Lagrangian multipliers of the above optimization problem.   


The above equation is the multidimensional counterpart of the SML in which Typesetting:-mrow(Typesetting:-mi(plays the rule of β. The price of one unit of risk of type Typesetting:-mrow(Typesetting:-mi( (captured by the Typesetting:-mrow(Typesetting:-mi(th factor) is Typesetting:-mrow(Typesetting:-mi(To reinforce this interpretation let us see what is Typesetting:-mrow(Typesetting:-mi( If a risk free asset exists in this market, and its rate of return is RTypesetting:-mrow(Typesetting:-mi(  then Typesetting:-msub(Typesetting:-mi(=0 for every Typesetting:-mrow(Typesetting:-mi( and thus 


Typesetting:-mrow(Typesetting:-mo(Typesetting:-munderover(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mo( . 

Now it is more apparent that the rate of return on security Typesetting:-mrow(Typesetting:-mi( is the risk free rate plus the compensation ofTypesetting:-mrow(Typesetting:-mo(per one unit of risk Typesetting:-mrow(Typesetting:-mi( . That is for each unit of factor Typesetting:-mrow(Typesetting:-mi( that is embedded in security Typesetting:-mrow(Typesetting:-mi( the compensation in terms of rate of return is Typesetting:-mrow(Typesetting:-mi(hence for Typesetting:-mrow(Typesetting:-mi(units of factor Typesetting:-mrow(Typesetting:-mi( the compensation is Typesetting:-mrow(Typesetting:-mi( and the total compensation is  Typesetting:-munderover(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mo( 

Similarly we can present the other Typesetting:-mrow(Typesetting:-mi( in terms of portfolios that are influenced only by one factor. These are portfolios that have risk profiles presented by a vector in the neutral basis. Let us denote by Typesetting:-mrow(Typesetting:-mi( the expected rate of return on a portfolio with a  risk profile of (0,...,1,0,...0) where 1 stands at the Typesetting:-mrow(Typesetting:-mi(th position. That is a portfolio that is sensitive only to the risk of the Typesetting:-mrow(Typesetting:-mi(th factor, i.e., Typesetting:-mrow(Typesetting:-mi( for all Typesetting:-mrow(Typesetting:-mi( except for Typesetting:-mrow(Typesetting:-mi(  Typesetting:-mrow(Typesetting:-mi( Hence for such a portfolio the equation


implies that 


We therefore can state that

Typesetting:-mrow(Typesetting:-mi(  for every i

which of course resembles the SML where Typesetting:-mrow(Typesetting:-mi(plays the rule of the rate of return on the market portfolio.  


5.4.1 The Lagrangian and the APT main result 


The Lagrangian of the optimization problem  


s.t.  Typesetting:-mrow(Typesetting:-mi( for  j=1,...,s






In chapter 1 we investigated the first order conditions for such (convex) optimization problems. We also mentioned that the optimal Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( correspond to a saddle point of the Lagrangian and hence that  




It is also easy to show from first principals that the optimal value of the optimization problem above equals to Typesetting:-mrow(Typesetting:-mi( Hence if no arbitrage opportunities exists 




The APT main result is a consequence of the first order condition of Typesetting:-mrow(Typesetting:-mi( and of the optimal value of this problem being zero.  This result is can be obtained applying virtually the same arguments (and geometric interpretation) that we used in Chapter 1 to prove the first order conditions for constrained optimization.  


For a fixed λ the first order condition of Typesetting:-mrow(Typesetting:-mi( are Typesetting:-mrow(Typesetting:-mi(which implies that   


Substituting Typesetting:-mrow(Typesetting:-mo(Typesetting:-munderover(Typesetting:-mo(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(for Typesetting:-mrow(Typesetting:-mi(in Typesetting:-mrow(Typesetting:-mi(indeed yields that Typesetting:-mrow(Typesetting:-mi(Thus  


if and only if  



and consequently there are no arbitrage opportunities in the market if and only if the above equality is satisfied. 



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