On Regularized Optimal Execution Problems and Their Singular Limits

Abstract

We investigate the portfolio execution problem under a framework in which volatility and liquidity are both uncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both of them. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law to relate it to the agent's turnover rate. We first analyse the regularized setting, in which the admissible strategies do not ensure complete execution of the initial inventory. We prove the existence and uniqueness of a continuous and bounded viscosity solution of the Hamilton–Jacobi–Bellman equation, whence we obtain a characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm. Then, we analyse the constrained problem, in which admissible strategies must guarantee complete execution to the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singular limit of the regularized counterparts.

Keywords:

• Optimal execution
• illiquid markets
• stochastic volatility
• stochastic liquidity
• viscosity solutions

Acknowledgements

We also thanks an anonymous reviewer and an associate editor for their helpful comments.

Disclosure statement

The authors declare that they have no conflict of interest.

Notes

1 As put in Bouchaud (2009), ‘price impact refers to the correlation between an incoming order (to buy or to sell) and the subsequent price change’.

Additional information

Funding

This study was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance code 001. MOS was partially supported by CNPq grant # 310293/2018-9.