Optimal Trading Execution with Nonlinear Market Impact: An Alternative Solution Method

TL;DR

This paper extends Almgren's optimal trade execution model by introducing a solution method that handles initial positive asset holdings, revealing conditions for no-trade periods and the influence of initial activity on execution timing.

Contribution

It proposes a new solution approach for nonlinear market impact models that accommodates initial positive asset holdings, unlike previous assumptions.

Findings

Optimal execution may involve no trading at the start under certain conditions.

Initial asset holdings influence the optimal trading schedule.

Additional parametric restrictions are necessary for no-trade solutions.

Abstract

We consider the optimal trade execution strategies for a large portfolio of single stocks proposed by Almgren (2003). This framework accounts for a nonlinear impact of trades on average market prices. The results of Almgren (2003) are based on the assumption that no shares of assets per unit of time are trade at the beginning of the period. We propose a general solution method that accomodates the case of a positive stock of assets in the initial period. Our findings are twofold. First of all, we show that the problem admits a solution with no trading in the opening period only if additional parametric restrictions are imposed. Second, with positive asset holdings in the initial period, the optimal execution time depends on trading activity at the beginning of the planning period.