Asset Pricing John H. Cochrane June 12, 2000

SECTION 3.2 RISK NEUTRAL PROBABILITIES
I denote the price p(x) to emphasize that it is the price of the payoff x. Where the payoff
in question is clear, I suppress the (x). I like to think of equation (3.48) as a happy-meal
theorem: the price of a happy meal (in a frictionless market) should be the same as the price
of one hamburger, one small fries, one small drink and the toy.
It is easier to take expectations rather than sum over states. To this end, multiply and
divide the bundling equation (3.48) by probabilities,
p(x) = X
µ
pc(s)
π(s) x(s)
π(s)
s
where π(s) is the probability that state s occurs. Then define m as the ratio of contingent
claim price to probability
m(s) = pc(s)
π(s) .
Now we can write the bundling equation as an expectation,
p = X
π(s)m(s)x(s) =E(mx).
s
Thus, in a complete market, the stochastic discount factor m in p = E(mx) exists, and it
is just a set of contingent claims prices, scaled by probabilities. As a result of this interpretation,
the combination of discount factor and probability is sometimes called a state-price
density.
The multiplication and division by probabilities seems very artificial in this finite-state
context. In general, we posit states of nature ω that can take continuous (uncountably infinite)
values in a space Ω. In this case, the sums become integrals, and we have to use some measure
to integrate over Ω. Thus, scaling contingent claims prices by some probability-like object is
unavoidable.
3.2 Risk neutral probabilities
I interpret the discount factor m as a transformation to risk-neutral probabilities such that
p = E ∗ (x)/R f .
Another common transformation of p = E(mx) results in “risk-neutral” probabilities.
Define
π ∗ (s) ≡ R f m(s)π(s) =R f pc(s)
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