Optimal Trade Execution Problem Based on Hamilton-Jacobi-Bellman Equations

DOI: https://doi.org/10.62517/jse.202611312

Author(s)

Yan Zhang*, Peng Li

Affiliation(s)

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan, China *Corresponding Author

Abstract

The optimal trade execution problem in financial markets is investigated within a risk-return trade-off perspective evaluated at the initial time. A linear-quadratic stochastic optimal control formulation is obtained by introducing the Lagrange multiplier method, from which the corresponding nonlinear Hamilton-Jacobi-Bellman equation is derived via the dynamic programming principle. To solve this equation, a semi-Lagrangian numerical scheme is employed. Its implementation offers considerable flexibility with respect to the particular specification of the price impact function. Under a suitable comparison principle, the theoretical convergence of this numerical scheme to the viscosity solution of the HJB equation follows. Numerical experiments are performed to recover the corresponding efficient trading frontiers and to analyze the optimal execution strategies. In particular, a sensitivity analysis is performed with respect to the market volatility parameter, for which four distinct values are considered, namely σ = 0.5, 1.0, 1.5, and 2.0. As the volatility increases, the efficient frontier shifts to the right and downward, indicating that risk rises for a given level of expected return, or alternatively, that expected return falls for a given level of risk. When the volatility tends toward zero, the efficient frontier shrinks to a single risk-free point, which helps confirm the limiting consistency of the proposed numerical algorithm. In addition, under some parameter settings, distinct trading strategies may generate nearly identical risk-return frontiers, a finding that sheds new light on the selection of practical trading strategies.

Keywords

Optimal Trade Execution; Mean-variance; HJB Equation; Semi-Lagrangian Method; Efficient Frontier

References

[1] Almgren, R., & Chriss, N. (2000). Optimal execution of portfolio transactions. Journal of Risk, 3(Winter), 5-39. [2] Bertsimas, D., & Lo, A. (1998). Optimal control of execution costs. Journal of Financial Markets, 1, 1-50. [3] Zhou, X. Y., & Li, D. (2000). Continuous time mean variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19-33. [4] Na, A. S., & Wan, J. W. L. (2024). Residual U-net with Self-Attention to Solve Multi-Agent Time-Consistent Optimal Trade Execution. arXiv preprint arXiv:2312.09353. [5] Barles, G., & Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear equations. Asymptotic Analysis, 4, 271-283. [6] Barles, G. (1997). Convergence of numerical schemes for degenerate parabolic equations arising in finance. In Numerical Methods in Finance. Cambridge University Press. [7] Palmari, G. (2024). Optimal execution with deterministically time varying liquidity: well posedness and price manipulation. arXiv:2410.04867. [8] Bank, P., Cartea, Á., & Körber, L. (2025). Optimal Execution and Speculation with Trade Signals. Fields Institute Talk, May 2025. [9] Huberman, G., & Stanzl, W. (2004). Price manipulation and quasi-arbitrage. Econometrica, 72, 1247-1275. [10]Anonymous. (2024). Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems. Finance and Stochastics, 28, 813–863. [11]Schied, A., & Schoeneborn, T. (2009). Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets. Finance and Stochastics, 13, 181-204. [12]Vath, V. L., Mnif, M., & Pham, H. (2007). A model of optimal portfolio selection under liquidity risk and price impact. Finance and Stochastics, 11, 51-90. [13]Lorenz, J., & Almgren, R. (2007). Adaptive arrival price. In Algorithmic Trading III: Precision, Control, Execution. Institutional Investor Journals. [14]Engle, R., & Ferstenberg, R. (2007). Execution risk. Journal of Trading, 2(2), 10-20. [15]Forsyth, P. A., Kennedy, J. S., Tse, S. T., & Windcliff, H. (2009). Optimal trade execution: a mean quadratic variation approach. Quantitative Finance (submitted).