An optimal execution problem with market impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales; otherwise the value function is continuous. Moreover, we show the semigroup property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We present some examples where the form of the optimal strategy changes completely, depending on the amount of the trader’s security holdings, and where optimal strategies in the Black–Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.

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