We study an optimal execution problem with uncertain market impact to derive a more realistic market model. First, we construct a discrete-time model as a value function for optimal execution. Market impact is formulated as the product of a deterministic part increasing with execution volume and a positive stochastic noise part. Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by a stochastic control problem with a Levy process. We show that noise in market impact will cause a risk-neutral trader to underestimate the impact cost. We study typical examples under a log-linear/quadratic market impact function with Gamma-distributed noise.
