In a market where there are no arbitrage opportunities, the following question can be posed: Given the profile of a certain cash flow in the future, what should its price be? Only in markets with no arbitrage opportunities can such a question make sense. As noted above, if arbitrage opportunities exist there may be more than one price assigned to a portfolio, as the law of one price does not hold. Moreover, one would not bother buying any security if one can make infinite amounts of money with no risk and no initial investment.

The question posed, however, may not have a unique solution. The absence of a unique solution may occur in markets termed incomplete markets . An incomplete market is a market where, given a cash flow profile, there might not be a portfolio that generates this cash flow. The appendix elaborates more on this point. The market in our example is complete.

We thus return the price of security 2 to its original value of $105 so that the no-arbitrage condition holds, and accordingly redefine the array Cash .

> Cash:=[Cost=60*S1+105*S2+100*SF,IncomeSt1=50*S1+90*S2+110*SF,IncomeSt2=60*S1+110*S2+110*SF,IncomeSt3=80*S1+130*S2+110*SF];

There are three "special'' cash flows in this market, sometimes also referred to as elementary cash flows. These can be priced now. The meaning of these cash flows is the topic of the next section. The first cash flow is ( ), i.e., in the next time period receive $1 in state 1, and $0 in states 2 and 3. In order to price this cash flow, we use our minimization problem, as before:

> simplex[minimize](60*S1+105*S2+100*SF,{50*S1+90*S2+110*SF>=1,60*S1+110*S2+110*SF>=0,80*S1+130*S2+110*SF>=0});

The optimal solution of the minimization problem,

,

is the portfolio we seek. It stipulates the combination of the three securities to be purchased now in order to generate the first elementary cash flow in the next time period. Substituting the optimal solution of the minimization problem into Cash (as always) verifies the solution and allows the cost to be calculated.

> subs(S1=1/10,S2=-1/10,SF=1/22,Cash);

If units of security 1 are purchased, units of security 2 are shorted, and units of the bond are purchased, the resultant portfolio will generate the first elementary cash flow. This portfolio will cost $ now. In the same manner, the prices of the other two elementary cash flows, ( ) and then ( ), are calculated and substituted in Cash :

> simplex[minimize](60*S1+105*S2+100*SF,{50*S1+90*S2+110*SF>=0,60*S1+110*S2+110*SF>=1,80*S1+130*S2+110*SF>=0});

> subs(S1=-1/5,S2=3/20,SF=-7/220,Cash);

> simplex[minimize](60*S1+105*S2+100*SF,{50*S1+90*S2+110*SF>=0,60*S1+110*S2+110*SF>=0,80*S1+130*S2+110*SF>=1});

> subs(S1=1/10,S2=-1/20,SF=-1/220,Cash);

We now know the portfolio combinations of securities 1 and 2, and of the bond, to purchase in the current time period in order to generate each of the three elementary cash flows in the next time period. We also know what these portfolios cost. These results are summarized in Table 1.2.

Table 1.2: Three Elementary Cash Flows

In fact, now that we have the prices of these elementary cash flows, it is possible to value any cash flow in this market in a very simple way. Consider the cash flow ( ) and suppose we can find three portfolios: portfolio denoted by with a price of that pays $ if state occurs and zero otherwise, for . In this case the price of the cash flow ( ) must equal (by the replication arguments) . However, the cash flow from is actually times the cash flow from the elementary cash flow. Consequently, portfolio that produces $ contingent on state is equivalent to buying times the portfolio producing $1 contingent on state . Portfolios producing the elementary cash flows will be referred to as elementary portfolios and will be denoted by , . In our example the compositions of the elementary portfolios are summarized in Table 1.3.

Table 1.3: The Portfolios Generating the Elementary Cash Flows

Indeed, we already know the prices of the elementary portfolios. In our example the portfolio producing $1 contingent on state 1 costs $ . Thus the portfolio producing $ contingent on state 1 will cost $ and the cost of the cash flow ( ) is simply

.

(1.3)

The above argument shows that the cash flow ( ) is generated by buying times the portfolio , times the portfolio , and times the portfolio ; i.e., the required portfolio is thus

.

(1.4)

Let us now apply this valuation technique to an example and compare it to our first valuation method. Consider the cash flow ( ). Based on equation (1.3) its price should be

> (1/22)*100 -(25/44)*20 +(13/44)*45;

This result can be confirmed by solving the optimization problem that finds the least-cost portfolio producing this cash flow:

> simplex[minimize](60*S1+105*S2+100*SF,{50*S1+90*S2+110*SF>=100,60*S1+110*S2+110*SF>=-20,80*S1+130*S2+110*SF>=45});

Substitution of the solution (the portfolio) in the array Cash results in the cost and the cash flows of the portfolio.

> subs(S1=37/2,S2=-61/4,SF=219/44,Cash);

Alternatively, confirmation can be carried out without the need to solve an optimization problem. Since we already know the units of securities 1 and 2 and the bond in each of the elementary portfolios, we can find the portfolio producing this cash flow utilizing equation (1.4). This is done by summing the units of security 1 in (follow the first row of Table 1.3) and multiplying it by for .

> 100*(1/10) -20*(-1/5) +45*(1/10);

The result of the above calculation is the number of units of security 1 in the portfolio producing the cash flow ( ). We then repeat this for security 2

> 100*(-1/10)-20*(3/20) +45*(-1/20);

and finally for the bond

> 100*(1/22) -20*(-7/220) +45*(-1/220);

We see that the two methods produce the same result. The prices of the portfolios producing the elementary cash flows can be interpreted as discount factors. This is elaborated on in the next section from a different perspective.