Asset Pricing John H. Cochrane June 12, 2000

where
CHAPTER 3 CONTINGENT CLAIMS MARKETS
R f ≡ 1/ X pc(s) =1/E(m).
The π ∗ (s) are positive, less than or equal to one and sum to one, so they are a legitimate set
of probabilities. Then we can rewrite the asset pricing formula as
p(x) = X
s
pc(s)x(s) = 1
Rf X
∗ E
π (s)x(s) = ∗ (x)
Rf .
I use the notation E∗ to remind us that the expectation uses the risk neutral probabilities π∗ instead of the real probabilities π.
Thus, we can think of asset pricing as if agents are all risk neutral, but with probabilities
π∗ in the place of the true probabilities π. The probabilities π∗ gives greater weight to states
with higher than average marginal utility m.
There is something very deep in this idea: risk aversion is equivalent to paying more
attention to unpleasant states, relative to their actual probability of occurrence. People who
report high subjective probabilities of unpleasant events like plane crashes may not have
irrational expectations, they may simply be reporting the risk neutral probabilities or the
product m × π. This product is after all the most important piece of information for many
decisions: pay a lot of attention to contingencies that are either highly probable or that are
improbable but have disastrous consequences.
The transformation from actual to risk-neutral probabilities is given by
π ∗ (s) = m(s)
E(m) π(s).
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 π∗ . The risk-neutral probability representation
of asset pricing is quite common, especially in derivative pricing where the results are
independent of risk adjustments.
The risk-neutral representation is particularly popular in continuous time diffusion processes,
because we can adjust only the means, leaving the covariances alone. In discrete time,
changing the probabilities typically changes first and second moments. Suppose we start with
a process for prices and discount factor
dp
p = µp dt + σ p dz
dΛ
Λ = µΛ dt + σ Λ dz.
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