Option-Implied Currency Risk Premia - Princeton University

√
δ
requires loadings to range from
δ ∗
√
= 0.81 (high interest rate currencies) to
δ
δ ∗
= 1.16 (low interest rate
currencies), when measured as a fraction of the loading of the reference currency (δ ∗ ). 17 Our estimates of global
factor loadings based on option data fall into a comparable, but slightly broader range.
The second stage of the calibration relaxes the assumption that loadings are constant (Specifications I and
III), allowing them to depend on the contemporaneous interest rate differential, ξt i = ξ i − Ψ t · (r
t,t+1 i − t,t+1) rUS .
We find strong evidence that Ψ t is positive on average, indicating that currencies with higher prevailing interest
rates tend to have lower global factor loadings. The estimate of the mean value of Ψ t is 0.42 (t-stat: 13.09) for
the option set including the high-low cross rates and X/USD currency pairs, and increases to 2.37 (t-stat: 62.17)
as the option set is expanded to include the full cross-section of 45 G10 cross rates (Table I, Panel B). Aside from
this conditional variation in loadings, we observe a strong negative unconditional cross-sectional relationship
between the ξ i and the mean interest rate differential relative to the U.S. (Table I, Panel A; Figure 2, top left
panel). Both of these dimensions of variation are consistent with the preferred (“unrestricted”) specification in
Lustig, et al. (2011). We plot the time series of global factor loadings from Specification I for two high interest
rate currencies (AUD, NOK) and two low interest rate currencies (CHF, JPY) in the top panel of Figure 1.
Figure 2 summarizes the unconditional, cross-sectional relation between the global factor loading differentials
and various data quantities. The top two panels consider the relation between the loading differentials and
interest rate differentials (left panel), as well as, the realized currency excess return (right panel). Recall that neither
of these two quantities was used as a target in the model calibration, which is designed to match FX option
prices. The top left panel indicates a strong negative relation between the mean loading differential and the mean
one-month interest rate differential. We find that this unconditional relation is driven primarily by the negative
relation between the estimates of ξ i and the interest rate differentials, rather than the estimate of Ψ t . This is
consistent with the link between loadings and interest rate differentials implied by our model of pricing kernel
dynamics, (14). Similarly, we observe a negative relation between the global factor loadings recovered from
exchange rate options and the mean realized currency pair excess returns. Likewise, this is consistent with the
model-predicted relation between currency risk premia and loading differentials, (9), suggesting that exchange
rate options carry information that is useful for explaining currency excess returns. The two bottom panels examine
the link between the calibrated global factor loadings and option-implied exchange rate moments (volatility
and skewness), both of which are used in the model calibration. The model roughly predicts that: (a) optionimplied
volatility is a function of the absolute loading differential (21a); and, (b) option-implied skewness is an
17 The “restricted” specification in Lustig, et al. (2011) fixes the sensitivity of the global loading to the country-specific state variable,
Yt i , at zero by setting κ = 0. Our estimates of the global loading coefficient, ξ i , map into √ δ i under their notation, with the added
normalization that we fix the loading of the U.S. dollar at unity.
22