Hedging nontradable risks with transaction costs and price impact

A risk‐averse agent hedges her exposure to a nontradable 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 nontradable risk factor is nonlinear, 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 is nonlinear, 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.