Asset Pricing John H. Cochrane June 12, 2000

Chapter 3. Contingent Claims Markets
Our firsttaskistounderstandthep = E(mx) representation a little more deeply. In this
chapter I introduce a very simple market structure, contingent claims. This leads us to an
inner product interpretation of p = E(mx) which allows an intuitive visual representation
of most of the theorems. We see that discount factors exist, are positive, and the pricing
function is linear, just starting from prices and payoffs in a complete market, without any
utility functions. The next chapter shows that these properties can be built up in incomplete
markets as well.
3.1 Contingent claims
I describe contingent claims. I interpret the stochastic discount factor m as contingent
claims prices divided by probabilities, and p = E(mx) as a bundling of contingent claims.
Suppose that one of S possible states of nature can occur tomorrow, i.e. specialize to a
finite-dimensional state space. Denote the individual states by s. For example, we might have
S =2and s = rain or s = shine.
A contingent claim is a security that pays one dollar (or one unit of the consumption
good) in one state s only tomorrow. pc(s) is the price today of the contingent claim. I write
pc to specify that it is the price of a contingent claim and (s) to denote in which state s the
claim pays off.
In a complete market investors can buy any contingent claim. They don’t necessarily have
to be faced with explicit contingent claims; they just need enough other securities to span
or synthesize all contingent claims. For example, if the possible states of nature are (rain,
shine), one can span or synthesize any contingent claim or portfolio achieved by combining
contingent claims by forming portfolios of a security that pays 2 dollars if it rains and one if
it shines, or x1 =(2, 1), and a riskfree security whose payoff pattern is x2 =(1, 1).
Now, we are on a hunt for discount factors, and the central point is:
If there are complete contingent claims, a discount factor exists, and it is equal to the
contingent claim price divided by probabilities.
Let x(s) denote an asset’s payoff in state of nature s. We can think of the asset as a
bundle of contingent claims—x(1) contingent claims to state 1, x(2) claims to state 2, etc.
The asset’s price must then equal the value of the contingent claims of which it is a bundle,
p(x) = X
pc(s)x(s). (48)
s
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