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Logarithmic regret in the ergodic Avellaneda-Stoikov market making model

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  • Jialun Cao
  • David v{S}iv{s}ka
  • Lukasz Szpruch
  • Tanut Treetanthiploet

Abstract

We analyse the regret arising from learning the price sensitivity parameter $\kappa$ of liquidity takers in the ergodic version of the Avellaneda-Stoikov market making model. We show that a learning algorithm based on a regularised maximum-likelihood estimator for the parameter achieves the regret upper bound of order $\ln^2 T$ in expectation. To obtain the result we need two key ingredients. The first are tight upper bounds on the derivative of the ergodic constant in the Hamilton-Jacobi-Bellman (HJB) equation with respect to $\kappa$. The second is the learning rate of the maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiment confirms the convergence and the robustness of the proposed algorithm.

Suggested Citation

  • Jialun Cao & David v{S}iv{s}ka & Lukasz Szpruch & Tanut Treetanthiploet, 2024. "Logarithmic regret in the ergodic Avellaneda-Stoikov market making model," Papers 2409.02025, arXiv.org.
    • Handle: RePEc:arx:papers:2409.02025
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    References listed on IDEAS

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    1. Ben Hambly & Renyuan Xu & Huining Yang, 2020. "Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon," Papers 2011.10300, arXiv.org, revised Jun 2021.
    2. Olivier Guéant & Iuliia Manziuk, 2020. "Optimal control on graphs: existence, uniqueness, and long-term behavior," Post-Print hal-03252606, HAL.
    3. Álvaro Cartea & Sebastian Jaimungal, 2015. "Risk Metrics And Fine Tuning Of High-Frequency Trading Strategies," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 576-611, July.
    4. Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
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