Hedging Non-Tradable Risks with Transaction Costs and Price Impact

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.