Option-Implied Currency Risk Premia - Princeton University

where S ji
t is the currency exchange rate, specified as the currency I price of currency J, and M i t and M j t are
the pricing kernels. If markets are complete, this restriction pins down the exchange rate, S ji
t , as the ratio of the
unique pricing kernels in the two economies. If markets are incomplete, it places an additional restriction on the
dynamics of any candidate stochastic discount factors, such that exchange rates and exchange rate options can be
viewed as helping complete markets. Consequently, the log currency return can be written as:
s ji
t+1 − sji t =
( )
m j t+1 − mj t
= −α j t + αi t −
− ( m i t+1 − m i )
t
(
ξ j t − ξi t
)
· L g Z t
− L j + L i Y j Y i
t t
(7)
We define the currency risk premium for currency pair J/I as the difference between the log expected return
on the currency adjusted for the interest rate differential, following Bakshi, et al. (2008):
λ ji
t ≡ ln E P t
[ ]
S
ji (
)
t+1
S ji + y j t,t+1 − yi t,t+1
t
(8)
To first order, this expression corresponds to the return on a zero-investment position which is long currency, J,
and is funded in currency, I. 8
This is an attractive feature because it implies that this model-implied quantity
can be aggregated using portfolio weights to produce a portfolio-level risk premium (e.g. for the HML F X and
“dollar carry” factor mimicking portfolios studied by Lustig, et al. (2011, 2013)). 9 Substituting in the expressions
for the risk-free rates, and using the cumulant generating function to re-write the conditional expectation we
8 The expected simple return on a long-short currency position is given by:
E P t
[
exp
( )
y j t,t+1
(
· Sji t+1
− exp yt,t+1) ] (
i = exp y j
S ji
t,t+1 + ln Et
P
t
≈
ln E P t
[ ]
S
ji
t+1
+ y j
S ji t,t+1 − yt,t+1
i
t
[ ])
S
ji
( )
t+1
− exp y i
S ji
t,t+1
t
where the second expression follows from a first-order Taylor expansion of the exponential function. The final, approximate expression
is identical to the formula in Bakshi, et al. (2008), though the expression reported in their paper has a typographical error and is missing
the logarithm in front of the gross currency return.
9 An alternative
[
measure of the currency risk premium used by Backus, et al. (2001) and Lustig, et al. (2013) is the mean log excess
return, Et
P s
ji
t+1 − ] ( sji t + y
j
t,t+1 − t,t+1) yi . The disadvantage of this measure is that it cannot be aggregated linearly using portfolio
weights to produce model-implied estimates of portfolio risk premia. The log risk premium can be expressed as a series expansion in
terms of the cumulants of the time-changed Lévy increments as:
E P t
[ ] (
)
s ji
t+1 − s ji
t + y j t,t+1 − yt,t+1
i
=
∞∑
(−1) k ·
k=2
( (ξ )
i k ( )
t − ξ
j k
)
t · κ k L g + κ k
Z L i t Y
t
i
k!
− κ k L j Y j t
10