Optimal Liquidation of Perpetual Contracts

An agent holds a position in a perpetual contract with payoff function $\psi$ and attempts to liquidate the position while managing transaction costs, inventory risk, and funding rate payments. By solving the agent's stochastic control problem we obtain a closed-form expression for the optimal trading strategy when the payoff function is given by $\psi(s) = s$. When the payoff function is non-linear we provide approximations to the optimal strategy which apply when the funding rate parameter is small or when the length of the trading interval is small. We further prove that when $\psi$ is non-linear, the short time approximation can be written in terms of the closed-form trading strategy corresponding to the case of the identity payoff function.