## Positive State Prices

If a complete set of elementary securities exists or can be constructed as a combination of existing securities, it is important that their prices be positive Otherwise there would be an arbitrage opportunity. To see this, suppose an elementary state security e% had a zero or negative price. That security would then present the possibility of obtaining something (a payoff of I if the state s occurs) for nonpositive cost, This is type B arbitrage So if elementary state securities actually exist or can be constructed as combinations of other securities, their prices must be positive to avoid arbitrage

Actually, the condition of no arbitrage possibility is equivalent to the existence of positive state prices as established by the following theorem:

Positive state prices theorem A set of positive state prices exists ij and only if there are no arbitrage opportunities

Proof: Suppose first that there are positive state prices. Then it is clear that no arbitrage is possible. To see this, suppose a security d can be constructed with d > 0. We have d = {dl, d2,. .., ds) with cls > 0 for each = 1, 2,. . . , 5. The price of d is P — which since t//v > 0 for all a, gives P > 0

Indeed P > 0 if d / 0 and P = 0 if d = 0 Hence there is no arbitrage possibility,

To prove the converse, we assume that there are no arbitrage opportunities, and we make use of the result on the portfolio choice problem of Section 9.7 This proof requires some additional assumptions. (A more general proof is outlined in Exercise 12,) We assume there is a portfolio such that Yl'!=\ > We assign positive probabilities ps, s = 1,2,..,, 5', to the states arbitrarily, with XlvLi P* ~ ^ anc^ we se^ec!- a strictly increasing utility function U. Since there is no arbitrage, there is, by the portfolio choice theorem of Section 9,8, a solution to the optimal portfolio choice problem We assume that the optimal payoff has ,,v* > 0, The necessary conditions (9.5) show that for any security cl with price P, where ..v* is the (random) payoff of the optima! portfolio and A > 0 is the Lagrange multiplier

If we expand this equation to show the details of the expected value operation, we find

s s where U'(x*)s is the value of U'(x*) in state .v Now we define

PsU\x*y

We see that i//v > 0 because ps >• 0, U'(x*yi > 0, and A. > 0 We also have

P = \frsds s=l showing that the V'/s are state prices They are all positive. 1

Note that the theorem says that such positive prices exist—it does not say that they are unique. If there are more states than securities, there may be many different ways to assign state prices that are consistent with the prices of the existing securities, The theorem only says that for one of these ways the state prices are positive.

Example 9.8 (The plain film venture) Consider again the original film venture There are three states, but only two securities: the venture itself and the riskless security, Hence state prices are not unique

We can find a set of positive state prices by using (9.12) and the values of the 0,-'s and X = I found in Example 9 5 (with W = 1), We have

1 202

These state prices can be used only to price combinations of the original two securities. They could not be applied, for example, to the purchase of residual rights, To check the price of the original venture we have P — 3 x .221 + ..3.38 — I, as it should be

Example 9.9 (Expanded film venture) Now consider the film venture with three available securities, as discussed in Example 9.6, which introduces residual rights Since there are three states and three securities, the state prices are unique Indeed we may find the state prices by setting the price of the three securities to 1, obtaining

This system has the solution

!//,=!, ^2 = 5, V'3 = g-Therefore the price of a security with payoff (d],d2,d^) is

9 10 RISK-NEUTRAL PRICING 251

You can compare this with the formula for P given at the end of Example 9.7 It is exactly the same.

Note also that these state prices, although different from those of the preceding example, give the same values for prices of securities that are combinations of just the two in the original film venture For example, the price of the basic venture itself is P = I + 3 = 1

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