Option-Implied Currency Risk Premia - Princeton University

Y j
t parameter can be interpreted as measuring the degree of similarity between the jump process and a Brownian
motion. The CGMY process nests compound Poisson jumps (−1 ≤ Y < 0), infinite-activity jumps with finite
variation (0 ≤ Y < 1), as well as, infinite-activity jumps with infinite variation (1 ≤ Y < 2). Specifically, we
assume that: (1) global jumps are one-sided, allowing only for positive shocks to marginal utility (M g t
= ∞); and,
(2) country-specific jumps are two-sided, capturing both positive and negative idiosyncratic shocks. Finally, we
use the (time-change) state-variables, Z t and Y i
t , to set the periodic volatility of the Lévy increments. Appendix
A discusses our specification in more detail.
Similar to Martin (2013), we rely on the cumulant generating functions (CGF) of the pricing kernel innovations,
k [u], to express quantities of interest such as yields, currency risk premia and and option prices. 7 The
cumulant generating function for the non-time-changed Lévy increments, L j t , are reported in Appendix A. To
derive the CGF for the corresponding time-changed increments we rely on Theorem 1 in Carr and Wu (2004). In
our setup, the time-change is controlled by pre-determined state-variables, Z t and Y i
t , which allow the model to
have non-identically distributed innovations over time. Unlike in a more typical stochastic volatility model (e.g.
Lustig, et al. (2011)), the time-change variables affect not only volatility, but also the higher order moments of
the pricing kernel, enabling the model to better match the empirical features of foreign exchange option data.
Theorem 1 of Carr and Wu (2004) states that for a generic time change, T , the cumulant generating function of
the time-changed Lévy process, L T , is given by k T [k L [u]], where k L [u] is the cumulant generating function of
the non-time-changed process and k T [u] is the cumulant generating function of the time-change. In our case, the
time-change variables are fixed within the measurement interval (i.e. they follow a degenerate stochastic process
with zero drift and volatility), such that:
[ ]
k L
g [u] = k Zt k
Z L
g [u]
t t
k L i
Y i
t
[u] = k Y i
t
k L i
t
[u]
[
]
= k L
g
t [u] · Z t
= k L i
t
[u] · Y i
t
(4a)
(4b)
Unless specifically noted with superscripts, cumulant generating functions are computed under the historical
(objective) measure, P.
7 Recall that the cumulant generating function of a random variable, ɛ t+1, is defined as follows:
k ɛ[u] = ln E t [exp (u · ɛ t+1)] =
∞∑ κ j · u j
where κ j, are the cumulants of the random variable, which can be computed by taking the j-th derivative of k ɛ[u] and evaluating the
resulting expression at zero. The cumulant generating function of the sum of two independent random variables is equal to the sum of
their cumulant generating functions.
j=1
j!
8