Asset Pricing John H. Cochrane June 12, 2000

State 2
Payoff
SECTION 3.5 STATE DIAGRAM AND PRICE FUNCTION
pc
Riskfree rate
Price = 0 (excess returns)
Price = 1 (returns)
State 1 contingent claim
Price = 2
Figure 7. Contingent claims prices (pc) and payoffs.
State 1 Payoff
gent claim price vector. We reasoned above that the price of the payoff x must be given by its
contingent claim value (3.48),
p(x) = X
pc(s)x(s). (50)
s
Interpreting pc and x as vectors, this means that the price is given by the inner product of the
contingent claim price and the payoff.
If two vectors are orthogonal – if they point out from the origin at right angles to each
other – then their inner product is zero. Therefore, the set of all zero price payoffs must lie
on a plane orthogonal to the contingent claims price vector, as shown in figure 7.
More generally, the inner product of two vectors x and pc equals the product of the magnitude
of the projection of x onto pc times the magnitude of pc. Using a dot to denote inner
product,
p(x) = X
pc(s)x(s) =pc · x = |pc|×|proj(x|pc)| = |pc|×|x|×cos(θ)
s
where |x| means the length of the vector x and θ is the angle between the vectors pc and
x. Since all payoffs on planes (such as the price planes in figure 7) that are perpendicular
to pc havethesameprojectionontopc, they must have the same price. (Only the price = 0
plane is, strictly speaking, orthogonal to pc. Lacking a better term, I’ve called the nonzero
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