A discrete-time benchmark tracking problem in two markets subject to random environments

• Published in: OR Spectrum Vol. 46; no. 4; pp. 1265 - 1294
• Main Authors: Jasso-Fuentes, Héctor, Salgado-Suárez, Gladys D.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2024; Springer Nature B.V
• Subjects: Bankruptcy; Benchmarks; Business and Management; Calculus of Variations and Optimal Control; Optimization; Derivatives; Deviation; Dynamic programming; Error analysis; Index funds; Investment strategy; Investments; Investors; Mathematics; Operations Research/Decision Theory; Original Article; Quadratic equations; Time measurement; Tracking problem; Valuation; Random environment; 90C40; Random walks; 91G15; 82B41; Benchmark tracking problems; 60K37; Decision processes; 93C30
• ISSN: 0171-6468; 1436-6304
• DOI: 10.1007/s00291-024-00767-x

In this manuscript, we study a benchmark tracking problem when prices evolve through a Binomial model with a random environment. The agent invests a given fund’s capital into different assets in a predetermined market to replicate at each stage of time a financial index or benchmark. To measure the actual deviation between the agent’s wealth and the current benchmark, we apply a deviation error expressed as a total sum of quadratic functions. We also assume the agent is obligated to change her/his investment between two markets when it is mandatory. This obligation happens when the fund’s wealth falls down a predetermined bankruptcy barrier. The dynamic programming method is then used to get optimal investment strategies that minimize the deviation error as well as to characterize the minimum deviation. We also apply the so-called potential function to analyze the influence of the environment on the prices. Numerical simulations are provided to illustrate our results.