Asset Pricing John H. Cochrane June 12, 2000

SECTION 1.5 DISCOUNT FACTORS IN CONTINUOUS TIME
Using the expanded version (1.32) in the basic equation (1.29), and dividing by pΛ to make
it pretty, we obtain an equivalent, slightly less compact but slightly more intuitive version,
0= D
· ¸
dΛ dp dΛ dp
dt + Et + + . (33)
p Λ p Λ p
(This formula only works when both Λ and p can never be zero. That is often enough the case
that this formula is useful. If not, multiply through by Λ and p and keep them in numerators.)
Applying the basic pricing equations (1.29) or (1.33) to a riskfree rate, defined as (1.25)
or (1.26), we obtain
r f
µ
dΛt
t dt = −Et
(34)
This equation is the obvious continuous time equivalent to
R f 1
t =
Et(mt+1) .
If a riskfree rate is not traded, we can use (1.34) to define a shadow riskfree rate or zero-beta
rate.
With this interpretation, we can rearrange equation (1.33) as
µ
dpt
Et +
pt
Dt
dt = r
pt
f
· ¸
dΛt dpt
t dt − Et . (35)
Λt pt
This is the obvious continuous-time analogue to
Λt
E(R) =R f − R f cov(m, R). (36)
The last term in (1.35) is the covariance of the return with the discount factor or marginal
utility. Since means are order dt, there is no difference between covariance and second moment
in the last term of (1.35). The interest rate component of the last term of (1.36) naturally
vanishes as the time interval gets short.
Ito’s lemma makes many transformations simple in continuous time. For example, the
nonlinear transformation between consumption and the discount factor led us to some tricky
approximations in discrete time. This transformation is easy in continuous time (diffusions
are locally normal, so it’s really the same trick). With Λt = e−δtu0 (ct) we have
dΛt = −δe −δt u 0 (ct)dt + e −δt u 00 (ct)dct + 1
2 e−δtu 000 (ct)dc 2 t
dΛt
Λt
= −δdt + ctu00 (ct)
u0 dct
(ct) ct
37
+ 1
2
c2 t u000 (ct)
u0 dc
(ct)
2 t
c2 t
(37)