  A risk-averse agent hedges her exposure to a non-tradable risk factor $U$
using a correlated traded asset $S$ and accounts for the impact of her trades
on both factors. The effect of the agent's trades on $U$ is referred to as
cross-impact. By solving the agent's stochastic control problem, we obtain a
closed-form expression for the optimal strategy when the agent holds a linear
position in $U$. When the exposure to the non-tradable risk factor $\psi(U_T)$
is non-linear, we provide an approximation to the optimal strategy in
closed-form, and prove that the value function is correctly approximated by
this strategy when cross-impact and risk-aversion are small. We further prove
that when $\psi(U_T)$ is non-linear, the approximate optimal strategy can be
written in terms of the optimal strategy for a linear exposure with the size of
the position changing dynamically according to the exposure's "Delta" under a
particular probability measure.
