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Optimal Tracking Portfolio with A Ratcheting Capital Benchmark

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  • Lijun Bo
  • Huafu Liao
  • Xiang Yu

Abstract

This paper studies the finite horizon portfolio management by optimally tracking a ratcheting capital benchmark process. It is assumed that the fund manager can dynamically inject capital into the portfolio account such that the total capital dominates a non-decreasing benchmark floor process at each intermediate time. The tracking problem is formulated to minimize the cost of accumulated capital injection. We first transform the original problem with floor constraints into an unconstrained control problem, however, under a running maximum cost. By identifying a controlled state process with reflection, the problem is further shown to be equivalent to an auxiliary problem, which leads to a nonlinear Hamilton-Jacobi-Bellman (HJB) equation with a Neumann boundary condition. By employing the dual transform, the probabilistic representation and some stochastic flow analysis, the existence of the unique classical solution to the HJB equation is established. The verification theorem is carefully proved, which gives the complete characterization of the feedback optimal portfolio. The application to market index tracking is also discussed when the index process is modeled by a geometric Brownian motion.

Suggested Citation

  • Lijun Bo & Huafu Liao & Xiang Yu, 2020. "Optimal Tracking Portfolio with A Ratcheting Capital Benchmark," Papers 2006.13661, arXiv.org, revised Apr 2021.
    • Handle: RePEc:arx:papers:2006.13661
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    References listed on IDEAS

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    1. Jun Sekine, 2012. "Long-term optimal portfolios with floor," Finance and Stochastics, Springer, vol. 16(3), pages 369-401, July.
    2. Erhan Bayraktar & Masahiko Egami, 2008. "An Analysis of Monotone Follower Problems for Diffusion Processes," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 336-350, May.
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    6. Weerasinghe, Ananda & Zhu, Chao, 2016. "Optimal inventory control with path-dependent cost criteria," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1585-1621.
    7. Di Giacinto, Marina & Federico, Salvatore & Gozzi, Fausto & Vigna, Elena, 2014. "Income drawdown option with minimum guarantee," European Journal of Operational Research, Elsevier, vol. 234(3), pages 610-624.
    8. El Karoui, Nicole & Jeanblanc, Monique & Lacoste, Vincent, 2005. "Optimal portfolio management with American capital guarantee," Journal of Economic Dynamics and Control, Elsevier, vol. 29(3), pages 449-468, March.
    9. Nicole El Karoui & Asma Meziou, 2006. "Constrained Optimization With Respect To Stochastic Dominance: Application To Portfolio Insurance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 103-117, January.
    10. Gaivoronski, Alexei A. & Krylov, Sergiy & van der Wijst, Nico, 2005. "Optimal portfolio selection and dynamic benchmark tracking," European Journal of Operational Research, Elsevier, vol. 163(1), pages 115-131, May.
    11. Strub, O. & Baumann, P., 2018. "Optimal construction and rebalancing of index-tracking portfolios," European Journal of Operational Research, Elsevier, vol. 264(1), pages 370-387.
    12. Shuoqing Deng & Xun Li & Huyen Pham & Xiang Yu, 2020. "Optimal Consumption with Reference to Past Spending Maximum," Papers 2006.07223, arXiv.org, revised Mar 2022.
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    Cited by:

    1. Li, Xun & Yu, Xiang & Zhang, Qinyi, 2023. "Optimal consumption and life insurance under shortfall aversion and a drawdown constraint," Insurance: Mathematics and Economics, Elsevier, vol. 108(C), pages 25-45.
    2. Héctor Jasso-Fuentes & Gladys D. Salgado-Suárez, 2024. "A discrete-time benchmark tracking problem in two markets subject to random environments," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(4), pages 1265-1294, December.
    3. Lijun Bo & Yijie Huang & Xiang Yu, 2023. "Stochastic control problems with state-reflections arising from relaxed benchmark tracking," Papers 2302.08302, arXiv.org, revised Apr 2024.
    4. Lijun Bo & Yijie Huang & Kaixin Yan & Xiang Yu, 2024. "Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint," Papers 2410.16611, arXiv.org.
    5. Chendi Ni & Yuying Li & Peter A. Forsyth, 2023. "Neural Network Approach to Portfolio Optimization with Leverage Constraints:a Case Study on High Inflation Investment," Papers 2304.05297, arXiv.org, revised May 2023.
    6. Bo, Lijun & Li, Tongqing & Yu, Xiang, 2022. "Centralized systemic risk control in the interbank system: Weak formulation and Gamma-convergence," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 622-654.
    7. Lijun Bo & Yijie Huang & Xiang Yu, 2023. "An extended Merton problem with relaxed benchmark tracking," Papers 2304.10802, arXiv.org, revised Jul 2024.
    8. Lijun Bo & Yijie Huang & Xiang Yu, 2023. "On optimal tracking portfolio in incomplete markets: The reinforcement learning approach," Papers 2311.14318, arXiv.org, revised Oct 2024.
    9. Lijun Bo & Tongqing Li & Xiang Yu, 2021. "Centralized systemic risk control in the interbank system: Weak formulation and Gamma-convergence," Papers 2106.09978, arXiv.org, revised May 2022.
    10. Xun Li & Xiang Yu & Qinyi Zhang, 2021. "Optimal consumption with loss aversion and reference to past spending maximum," Papers 2108.02648, arXiv.org, revised Mar 2024.
    11. van Staden, Pieter M. & Forsyth, Peter A. & Li, Yuying, 2024. "Across-time risk-aware strategies for outperforming a benchmark," European Journal of Operational Research, Elsevier, vol. 313(2), pages 776-800.

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