Building Up an Illiquid Stock Position Subject to Expected Fund Availability: Optimal Controls and Numerical Methods

• Published in: Applied mathematics & optimization Vol. 76; no. 3; pp. 501 - 533
• Main Authors: Lu, Xianggang, Yin, George, Zhang, Qing, Zhang, Caojin, Guo, Xianping
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2017; Springer Nature B.V
• Subjects: Approximation; Brownian motion; Calculus of Variations and Optimal Control; Optimization; Control; Control methods; Economic models; Lagrange multiplier; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Numerical analysis; Numerical and Computational Physics; Numerical methods; Optimal control; Optimization; Simulation; Stochastic processes; Systems Theory; Theoretical; Lagrange multiplier; 93E20; Constrained stochastic control; Numerical method; 91G60; 91G80
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9359-z

This paper is concerned with modeling, analysis, and numerical methods for stochastic optimal control of an illiquid stock position build-up. The stock price model is based on a geometric Brownian motion formulation, in which the drift is allowed to be purchase-rate dependent to capture the “price impact” of heavy share accumulation over time. The expected fund (or capital) availability has an upper bound. A Lagrange multiplier method is used to treat the constrained control problem. The stochastic control problem is analyzed and a verification theorem is developed. Although optimality is proved, a closed-form solution is virtually impossible to obtain. As a viable alternative, approximation schemes are developed, which consist of inner and outer approximations. The inner approximation is a numerical procedure for obtaining optimal strategies based on a fixed parameter of the Lagrange multiplier. The outer approximation is a stochastic approximation algorithm for obtaining the optimal Lagrange multiplier. Convergence analysis is provided for both the inner and outer approximations. Finally, numerical examples are provided to illustrate our results.