Asset Pricing John H. Cochrane June 12, 2000

price, p(x) > 0.
CHAPTER 4 THE DISCOUNT FACTOR
No-arbitrage says that you can’t get for free a portfolio that might pay off positively, but
will certainly never cost you anything. This definition is different from the colloquial use of
the word “arbitrage.” Most people use “arbitrage” to mean a violation of the law of one price
– a riskless way of buying something cheap and selling it for a higher price. “Arbitrages” here
might pay off, but then again they might not. The word “arbitrage” is also widely abused.
“Risk arbitrage” is a Wall Street oxymoron that means making specific kinds of bets.
An equivalent statement is that if one payoff dominates another, then its price must be
higher – if x ≥ y, then p(x) ≥ p(y) (Or, a bit more carefully but more long-windedly, if
x ≥ y almost surely and x>ywith positive probability, then p(x) >p(y). You can’t forget
that x and y are random variables.)
m>0 ⇒No-arbitrage
The absence of arbitrage opportunities is clearly a consequence of a positive discount
factor, and a positive discount factor naturally results from any sort of utility maximization.
Recall,
m(s) =β u0 [c(s)]
u 0 (c)
It is a sensible characterization of preferences that marginal utility is always positive. Few
people are so satiated that they will throw away money. Therefore, the marginal rate of
substitution is positive. The marginal rate of substitution is a random variable, so “positive”
means “positive in every state of nature” or “in every possible realization.”
Now, if contingent claims prices are all positive, a bundle of positive amounts of contingent
claims must also have a positive price, even in incomplete markets. A little more
formally,
> 0.
Theorem: p = E(mx) and m(s) > 0 imply no-arbitrage.
Proof: m>0; x ≥ 0 and there are some states where x>0. Thus, in some states
mx > 0 and in other states mx =0. Therefore E(mx) > 0. ¤
No arbitrage ⇒ m>0
Now we turn the observation around, which is again the hard and interesting part. As
the law of one price property guaranteed the existence of a discount factor m, no-arbitrage
guarantees the existence of a positive m.
The basic idea is pretty simple. No-arbitrage means that the prices of any payoff in the
positive orthant (except zero, but including the axes) must be strictly positive. The price =
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