Theoretical and Numerical Analysis of an Optimal Execution Problem with Uncertain Market Impact

TL;DR

This paper analyzes an optimal execution problem with uncertain market impact, establishing properties of the value function and examining how market impact noise affects cost assessment, with specific examples under certain impact models.

Contribution

It provides a detailed theoretical analysis of the value function's properties and explores the effects of market impact noise on cost estimation in optimal execution.

Findings

Value function is continuous and has the semigroup property.

Market impact noise leads to underestimation of impact costs.

Analysis includes examples with Gamma-distributed noise under specific impact functions.

Abstract

This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we show that the function is continuous and has the semigroup property, which is strongly related to the Hamilton-Jacobi-Bellman quasi-variational inequality. Moreover, we show that noise in market impact causes risk-neutral assessment to underestimate the impact cost. We also study typical examples under a log-linear/quadratic market impact function with Gamma-distributed noise.