On Regularized Optimal Execution Problems and Their Singular Limits. Issue 2 (4th March 2022)

• Record Type: Journal Article
• Authors: Souza, Max O.; Thamsten, Y.
• Is Part Of: Applied mathematical finance. Volume 29:Issue 2(2022)
• Page Start: 79
• Page End: 109
• Publication Date: 2022-03-04
• Subjects: Optimal execution -- illiquid markets -- stochastic volatility -- stochastic liquidity -- viscosity solutions; Business mathematics -- Periodicals; 650.0151
• DOI: 10.1080/1350486X.2022.2148115
• Languages: English
• ISSNs: 1350-486X

Abstract

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.