文档介绍：Ec2723, Asset Pricing I
Class Notes, Fall 2003
Complete Markets, plete Markets,
and the Stochastic Discount Factor
John Y. Campbell1
First draft: July 30, 2003
This version: September 15, 2003
1 Department of Economics, Littauer Center, Harvard University, Cambridge MA 02138, USA.
Email john_******@.
pletemarket
Consider a simple discrete-state model with states of nature s =1...
markets plete, that is, for each state s a contingent claim is available that
pays $1 in state s and nothing in any other state. Write the price of this contingent
claim as Pc(s).
Any other asset is defined by its payoff X (s) in state
XS
P (X)= Pc(s)X(s).
s=1
Multiply and divide by the probability of each state, π(s):
XS P (s) XS
P (X)= π(s) c X(s)= π(s)M(s)X(s)=E[MX],
π(s)
s=1 s=1
where M(s) is the ratio of state price to probability for state s,thestochastic discount
factor or SDF Any asset price is the expected product of the asset’s payoff and the
SDF.
Consider a riskless asset with payoff X(s)=1in every state. The price
XS
Pf = Pc(s)=E[M],
s=1
so the riskless interest rate
1 1
1+Rf = = .
Pf E[M]
Now define risk-neutral probabilities or pseudo-probabilities
M(s)
π(s)=(1+R )P (s)= π(s).
∗ f c E[M]
P
We have π∗(s) > 0 and s π∗(s)=1, so they can be interpreted as if they were
probabilities. We can rewrite the asset equation as
µ ¶ µ ¶
1 XS 1
P (X)= π(s)X(s)= E [X].
1+R ∗ 1+R ∗
f s=1 f
1
The price of any asset is the pseudo-expectation of its payoff, discounted at the riskless
interest rate.
Utility maximization and the SDF
Consider an investor with initial wealth Y and e Y (s). The investor’s
maximization problem is
XS
Max u(C0)+ βπ(s)u(C(s))
s=1
subject to
XS XS
C0 + Pc(s)C(s)=Y0 + Pc(s)Y (s).
s=1 s=1
Defining a Lagrange multiplier λ on the budget constraint, the first-order conditions
are
u0(C0)=λ
βπ(s)u0(C(s)) = λPc(s) for s =1...S.
Thus
P (s) βu (C(