Asset Pricing John H. Cochrane June 12, 2000

CHAPTER 4 THE DISCOUNT FACTOR
shocks that drive the original assets,
dΛ∗ Λ∗ = −rf µ
dt − µ + D
0
− rf Σ
p −1 σdz. (53)
where Σ ≡ σσ0 again is the covariance matrix of returns. You can easily check that this
equation solves
and
Et
µ
dp
+
p
D
p dt − rf µ ∗ dΛ
dt = −Et
Λ∗ Et
µ ∗ dΛ
= −r f dt,
Λ ∗
dp
p
or you can show that this is the only diffusion driven by dz, dt with these properties. If there
is a risk free rate r f
t
(54)
(also potentially time-varying), then that rate determines rf
t .Ifthereis
no risk free rate, (4.53) will price the risky assets for any arbitrary (or convenient) choice of
r f
t . As usual, this discount factor is not unique; Λ∗ plus orthogonal noise will also act as a
discount factor:
dΛ
Λ
dΛ∗
= + dw; E(dw) =0; E(dzdw) =0.
Λ∗ You can see that (4.53) is exactly analogous to the discrete time formula (4.52). (If you don’t
like answers popping out of hats like this, guess a solution of the form
dΛ
Λ = µ Λdt + σΛdz.
Then find µ Λ and σΛ to satisfy (4.54) for the riskfree and risky assets.)
4.4 Problems
1. Show that the law of one price loop implies that price is a linear function of payoff and
vice versa
2. Does the absence of arbitrage imply the law of one price? Does the law of one price
imply the absence of arbitrage? Answer directly using portfolio arguments, and indirectly
using the corresponding discount factors.
3. If the law of one price or absence of arbitrage hold in population, must they hold in a
sample drawn from that population?
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