Option-Implied Currency Risk Premia - Princeton University

coupon bond in country i; otherwise, it is the risk-neutral measure, Q i . In our discretized model, we assume the
interest rate is fixed over the life of the option at the level given by (5), such that the risk-forward and risk-neutral
measures coincide. Although we write the subsequent formulas under the risk-forward measure, these can be
interpreted as corresponding to risk-neutral quantities in the model. Finally, it is important to emphasize, that the
pricing measure depends on the home country of the investor, since investors in different countries have distinct
pricing kernels.
The τ-forward measure for an investor from country i is defined as follows:
dF i τ
dP
= M i t+τ
M i t
· exp ( y i t,t+τ · τ ) (18)
where P denotes the physical (historical) measure. The cumulant generating functions of the global and countryspecific
Lévy increments under the risk-forward measure, whose numeraire is the t + 1 zero-coupon bond, are:
k Fi
L g [u] = ln Et
Fi
Z t
(
=
k Fi
L
[u] =
i
Y
t
i
k L
g
t
k Fi [u] = k
L j L
j [u] · Y j
t
Y j
t
t
[ (
exp u · L
g
)] [ (
Z t = ln E
P
t exp y
i
t,t+1 + ( m i t+1 − m i )
t + u · L
g
[ ]
u − ξ
i
t − kL g
t
(
k L i
t
[u − 1] − k L i
t
[−1]
Z t
)]
[
−ξ
i
t
] ) · Z t (19a)
)
· Y i
t
(19b)
(19c)
Contrasting these expressions with the corresponding values under the objective measure, P, we see that the
change of measure results in: (1) a change in the distribution of the global factor dependent on country i’s loading
on the global factor, ξt; i (2) a change in the distribution of the local (reference) shock, L i ; and, (3) leaves the
Yt
i
distribution of foreign, country-specific shocks unchanged. To obtain the cumulant generating function for the
exchange rate at time t + 1 under the risk forward measure, F i , we substitute (7), into the definition of the CGF
to obtain:
k Fi
s ji
t
[u] =
(
) [( ) ]
s ji
t − α j t + αi t · u + k F i
L g ξt i − ξ j t · u · Z t + k F i
[u] · Y
t
L i t i + k F i
[−u] · Y j
t
L j t (20)
t
where each of the risk-forward CGFs can be evaluated using the formulas above. In order to substitute out the α i t
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