Asset Pricing John H. Cochrane June 12, 2000

CHAPTER 4 THE DISCOUNT FACTOR
negative, negating the assumption of no-arbitrage. ¤
The tough part comes if markets are incomplete. There are now many m’s that price
assets. Any m of the form m = x ∗ + ², with E(²x) =0will do. We want to show that at
least one of these is positive. But that one may not be x ∗ . Since the discount factors other
than x ∗ are not in the payoff space X, we can’t use the construction of the last argument,
since that construction may yield a payoff that is not in X, and hence to which we don’t
know how to assign a price. To handle this case, I adopt a different strategy of proof. (This
strategy is due to Ross 1978. Duffie 1992 has a more formal textbook treatment.) The basic
idea is another “to every plane there is a perpendicular line” argument, but applied to a space
that includes prices and payoffs together. As you can see, the price = 0 plane is a separating
hyperplane between the positive orthant and the negative payoffs, and the proof builds on this
idea.
Theorem: No arbitrage implies the existence of a strictly positive discount factor,
m>0,p= E(mx) ∀ x ∈ X.
Proof : Join (−p(x),x) together to form vectors in R S+1 . Call M the set of all
(−p(x), x) pairs,
M = {(−p(x),x); x ∈ X}
M is still a linear space: m1 ∈ M, m2 ∈ M ⇒ am1 + bm2 ∈ M. No-arbitrage
meansthatelementsofM can’t have all positive elements. If x is positive, −p(x)
must be negative. Thus, M is a hyperplane that only intersects the positive orthant
R S+1
+ at the point 0. We can then create a linear function F : RS+1 ⇒ R such that
F (−p, x) =0for (−p, x) ∈ M, andF (−p, x) > 0 for (−p, x) ∈ R S+1
+ except
the origin. Since we can represent any linear function by a perpendicular vector,
there is a vector (1,m) such that F (−p, x) =(1,m) · (−p, x) =−p + m · x or
−p + E(mx) using the second moment inner product. Finally, since F (−p, x) is
positive for (−p, x) > 0,mmust be positive.
¤
In a larger space than R S+1
+ , as generated by continuously valued random variables, the
separating hyperplane theorem assures us that there is a linear function that separates the two
convex sets M and the equivalent of R S+1
+ , and the Riesz representation theorem tells us that
we can represent F as an inner product with some vector by F (−p, x) =−p + m · x.
What the theorem does and does not say
The theorem says that a discount factor m>0 exists, but it does not say that m>0 is
unique. The left hand panel of Figure 12 illustrates the situation. Any m on the line through
x ∗ perpendicular to X also prices assets. Again, p = E[(m + ε)x] if E(εx) =0. All of
these discount factors that lie in the positive orthant are positive, and thus satisfy the theorem.
There are lots of them! In a complete market, m is unique, but not otherwise.
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