Asset Pricing John H. Cochrane June 12, 2000

The discount factor prices the assets,
Et
µ dp
p
SECTION 3.3 INVESTORS AGAIN
+ D
p dt − rf dt = −Et
µ dΛ
Λ
dp
= −σ
p
p σ Λ dt
In the “risk-neutral measure” we just increase the drift of each price process by its covariance
with the discount factor, and write a risk-neutral discount factor,
dp
p = ¡ µ p + σ p σ Λ¢ dt + σ p dz = µ p∗ dt + σ p dz
dΛ
Λ = µΛdt. Under this new set of probabilities, we can just write,
with E ∗ t (dp/p) =µ p∗ dt.
3.3 Investors again
E ∗ t
µ dp
p
+ D
p dt − rf dt =0
We look at investor’s first order conditions in a contingent claims market. The marginal
rate of substitution equals the discount factor and the contingent claim price ratio.
Though the focus of this chapter is on how to do without utility functions, It’s worth
looking at the investor’s first order conditions again in the contingent claim context. The
investor starts with a pile of initial wealth y and a state-contingent income y(s). He purchases
contingent claims to each possible state in the second period. His problem is then
max
{c,c(s)} u(c)+X βπ(s)u[c(s)] s.t. c + X
pc(s)c(s) =y + X
pc(s)y(s).
s
Introducing a Lagrange multiplier λ on the budget constraint, the first order conditions are
Eliminating the Lagrange multiplier λ,
s
u 0 (c) =λ
βπ(s)u 0 [c(s)] = λpc(s).
pc(s) =βπ(s) u0 [c(s)]
u 0 (c)
57
s