State Prices

A special form of security is one that has a payoff in only one state Indeed, we can define the S elementary state securities ex = {0,0,..,, 0, 1,0, ,0), where the I is the component s for s — 1,2, ,5 If such a security exists, we denote its price by

When a complete set of state securities exists (one for each state), it is easy to determine the price of any other security. The security d = (i/1, d2,. . . , ds) can be expressed as a combination of the elementary state securities as d = ELi^'V*. and hence by the linearity of pricing, the price of d must be s p = ]TVi//v. (9.10)

If the elementary state securities do not exist, it may be possible to construct them artificially by combining securities that do exist. For example, in a two-state world, if (1, 1) and (1, —1) exist, then one-half the sum of these two securities is equivalent to the first elementary state security (1, 0).

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