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【正文】 the contingent claims. ? p(x) =∑pc(s)x(s) () It is easier to take expectations rather than sum over states. ? p(x) = ∑? (s) (pc(s)/? (s)) x(s), where ? (s) is the probability that state s occurs. ? Then define m as the ratio of contingent claim price to probability， m(s) = pc(s)/? (s) Conclusion about discount factor ? Now we can write the bundling equation as an expectation, p = ∑?(s) m(s) x(s) = E(mx) ? If there are plete contingent claims, a discount factor exists, and it is equal to the contingent claim price divided by probabilities. Expand to infinite space ? In general, we posit states of nature ω that can take continuous (uncountably infinite) values in a space ?. In this case, the sums bee integrals, and we have to use some measure to integrate over ?. Thus, scaling contingent claims prices by some probabilitylike object is unavoidable. Risk neutral probabilities ? Define ? The ?*(s) are positive, less than or equal to one and sum to one, so they are a legitimate set of probabilities. ()()()( ) ( ) ( ) ( ) ( ) ,()( ) ( ) ( ) ( )()ssfpc sspc spc s m s s pc s E mmss s R m s sEm??? ? ????? ? ?? ? ???Then we can rewrite the asset pricing formula as: ? We use the notation E* to remind us that the expectation uses the risk neutral probabilities π *instead of the real probabilities π . ?*(s) = (m(s) / E(m)) ?(s) ? ?*gives greater weight to states with higher than average marginal utility m ? risk aversion is equivalent to paying more attention to unpleasant states, relative to their actual probability of occurrence. ? Application to report one’s reasonable subjective probabilities. ? We can also think of the discount factor m as the derivative or change of measure from the real probabilities π to the subjective probabilities ?* 連續(xù)時間 ?
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