where R* is the return on the log-optimal portfolio,
Isn't this a simple and easily remembered result? The formula looks very similar to the expression P — d/R that would hold in the case where d is deterministic. In the random case we just substitute R* for R and put an expected value in front, If d happens to be deterministic, this more general result reduces to the simple one because E(i //?*) = \/R.
Example 9,7 (Film variations) Suppose that a new security is proposed with payoffs that depend only on the possible outcomes of the film venture For example, one might propose an investment that paid back something even if the venture was a failure. A general security of this type will have payoffs d[,d2, and d\ corresponding to high success, moderate success, and failure, respectively. We can find the appropriate price of such a security by using the log-optimal portfolio that we calculated in Example 9 6 Note that we cannot use the simple log-optimal portfolio of the first film venture example, because that example only considered the film venture and the risk-free security, If a new security were a combination of those two, then we could use the simple log-optimal portfolio for pricing. But if the new security is a general one, we must use the log-optimal portfolio of the second example, since it includes a complete set of three securities for the three possibilities Any new security will be a combination of these three.
The log-optimal portfolio has the following return:
High Success Moderate Success Failure
These returns are calculated from the 0,'s found in the residual rights example. For example, under high success R* = — 1 0 x 3 -f 1.5 x 1.2 + .5 x 6 = 1.8 The value of a security with payoffs di ,d2, d3 is E(d/R*), which is tf1 d2 d3 p = 3— + 4— + 3 —
You can try this on the three securities we have used before; their prices should all turn out to be 1. For example, for the original venture, P — ,3-^j + ,4-jj = \ 4- ^ = 1,
We shall return to this log-optimal pricing equation in Chapter 15. For the moment we may regard it simply as a special version of the general pricing equation— the version obtained by using In ,.v as the utility function
Remember what is happening here. The prices of the original securities were used to find .v*. Now we use „v* to find those prices again, However, since pricing is linear, we can find the price of any security that is a linear combination of the original ones by the same formula.
What about a new security d that is not a linear combination of the original ones? We could enter it into the pricing equation as well, but the price obtained this way may not be correct. The formula is valid only for the securities used to derive it, or for a linear combination of those original securities
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