Option-Implied Currency Risk Premia - Princeton University

and α j t terms, we take advantage of the fact that: kFi [1] = s ji
s ji t + yt,t+1 i − yj t,t+1 .10 Finally, the exponential of the
t ( )
cumulant generating function, (20), evaluated at i · u, exp k Fi [i · u] , corresponds to the generalized Fourier
transform of the log currency return. From there, the Fourier transform can be inverted numerically to produce
prices for calls and puts, as in Carr and Madan (1999).
s ji
t
1.4.1 Option-implied moments
An attractive feature of our model, which we exploit in the empirical calibration, is that the cumulants of
the distribution are linear functions of the state variables, {Z t , Yt i , Y j }. For example, the second and third
cumulants under the risk-forward measure, F i are given by: 11
t
κ 2,Fi
s ji
t
κ 3,Fi
s ji
t
=
=
( ) 2 (
ξt i − ξ j t ·
( ) 3 (
ξt i − ξ j t ·
k L
g
t
k L
g
t
) ′′ [−ξ ] ( ) ′′
i
t · Zt + k L i
t
[−1] · Y
i
t +
) ′′′ [−ξ ] ( ) ′′′
i
t · Zt + k L i
t
[−1] · Y
i
t −
(
k L
j
t
(
k L
j
t
) ′′
[0] · Y
j
t
) ′′′
[0] · Y
j
t
(21a)
(21b)
Consequently, given a set of model parameters at time t, the state variables can be recovered using a crosssectional
regression of cumulants measured from contemporaneous exchange rate option data onto the modelimplied
coefficients. Although claims on cumulants are not traded, they can be readily reconstructed from option
prices. Specifically, using the insights from Breeden and Litzenberger (1978) and Bakshi, et al. (2003), we
compute option-implied swap rates for variance ( ˆV s
ji
t
) and skewness (Ŝs ji ). The corresponding empirical esti-
t
mates of the risk-forward cumulants can then be recovered from the definitions linking moments and cumulants,
ˆκ 2,Fi = ˆV
( ) 3
s ji s
ji, and ˆκ 3,Fi
2
=
t
t s ji Ŝsji
· ˆVs ji . With the prices of these claims in hand, the values of the state variables
t
t t
can be inferred directly by means of a cross-sectional linear regression, sidestepping more complicated filtering
procedures.
10 To obtain this result consider pricing an investment in currency J from the perspective of an investor in country i:
[
M i ( ) ] [
t+1
1 = E t · Sji t+1
· exp y j M i ( )
(
) ]
t+1
Mt
i S ji
t,t+1 = E t · exp y i
M i t,t+1 · Sji t+1
· exp y j
t
t
S ji
t,t+1 − yt,t+1
i
t
= E Fi
t
[
S
ji (
t+1
· exp y j
S ji
t,t+1 − yt,t+1) ]
i = exp
t
(k Fi
s ji
t
[1] + y j t,t+1 − y i t,t+1
11 The cumulants under the historical measure, P, are given by the same expressions, but with the derivatives of the consecutive cumulant
generating functions evaluated at zero, rather than {−ξ i , −1, 0}.
)
15