Optimal execution with dynamic risk adjustment

This article considers the problem of optimal liquidation of a position in a risky security quoted in a financial market, where price evolution are risky and trades have an impact on price as well as uncertainty in the filling orders. The problem is formulated as a continuous time stochastic optimal control problem aiming at maximising a generalised risk-adjusted profit and loss function. The expression of the risk adjustment is derived from the general theory of dynamic risk measures and is selected in the class of g-conditional risk measures. The resulting theoretical framework is nonclassical since the target function depends on backward components. We show that, under a quadratic specification of the driver of a backward stochastic differential equation, it is possible to find a closed form solution and an explicit expression of the optimal liquidation policies. In this way, it is immediate to quantify the impact of risk adjustment on the profit and loss and on the expression of the optimal liquidation policies.