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Optimal market making under partial information and numerical methods for impulse control games with applications

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  • Diego Zabaljauregui

Abstract

The topics treated in this thesis are inherently two-fold. The first part considers the problem of a market maker optimally setting bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. This stochastic control problem under partial information is solved by means of stochastic filtering, control and PDMPs theory. The value function is characterized as the unique continuous viscosity solution of its dynamic programming equation and numerically compared with its full information counterpart. The optimal full information spreads are shown to be biased when the exact market regime is unknown, as the market maker needs to adjust for additional regime uncertainty in terms of PnL sensitivity and observable order flow volatility. The second part deals with numerically solving nonzero-sum stochastic impulse control games. These offer a realistic and far-reaching modelling framework, but the difficulty in solving such problems has hindered their proliferation. A policy-iteration-type solver is proposed to solve an underlying system of quasi-variational inequalities, and it is validated numerically with reassuring results. Eventually, the focus is put on games with a symmetric structure and an improved algorithm is put forward. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. The algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise too challenging problems, and even some for which results go beyond the scope of the currently available theory.

Suggested Citation

  • Diego Zabaljauregui, 2020. "Optimal market making under partial information and numerical methods for impulse control games with applications," Papers 2009.06521, arXiv.org.
  • Handle: RePEc:arx:papers:2009.06521
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    References listed on IDEAS

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    1. Olivier Guéant & Charles-Albert Lehalle, 2015. "General Intensity Shapes In Optimal Liquidation," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 457-495, July.
    2. Erhan Bayraktar & Michael Ludkovski, 2014. "Liquidation In Limit Order Books With Controlled Intensity," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 627-650, October.
    3. Abel Cadenillas & Tahir Choulli & Michael Taksar & Lei Zhang, 2006. "Classical And Impulse Stochastic Control For The Optimization Of The Dividend And Risk Policies Of An Insurance Firm," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 181-202, January.
    4. Jean-Philippe Chancelier & Marouen Messaoud & Agnès Sulem, 2007. "A policy iteration algorithm for fixed point problems with nonexpansive operators," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 239-259, April.
    5. Olivier Gu'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Dealing with the Inventory Risk. A solution to the market making problem," Papers 1105.3115, arXiv.org, revised Aug 2012.
    6. René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 205-232, February.
    7. Ralf Korn, 1999. "Some applications of impulse control in mathematical finance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(3), pages 493-518, December.
    8. Pietro Fodra & Mauricio Labadie, 2012. "High-frequency market-making with inventory constraints and directional bets," Papers 1206.4810, arXiv.org.
    9. Olivier Gu'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Optimal Portfolio Liquidation with Limit Orders," Papers 1106.3279, arXiv.org, revised Jul 2012.
    10. Sylvain Delattre & Christian Y. Robert & Mathieu Rosenbaum, 2013. "Estimating the efficient price from the order flow: a Brownian Cox process approach," Papers 1301.3114, arXiv.org, revised Apr 2013.
    11. Matteo Basei, 2018. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Papers 1803.08166, arXiv.org, revised Mar 2019.
    12. Nicole Bäuerle & Ulrich Rieder, 2009. "MDP algorithms for portfolio optimization problems in pure jump markets," Finance and Stochastics, Springer, vol. 13(4), pages 591-611, September.
    13. Pietro Fodra & Mauricio Labadie, 2013. "High-frequency market-making for multi-dimensional Markov processes," Papers 1303.7177, arXiv.org, revised Apr 2013.
    14. Seydel, Roland C., 2009. "Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3719-3748, October.
    15. Ho, Thomas & Stoll, Hans R., 1981. "Optimal dealer pricing under transactions and return uncertainty," Journal of Financial Economics, Elsevier, vol. 9(1), pages 47-73, March.
    16. 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.
    17. Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
    18. Delattre, Sylvain & Robert, Christian Y. & Rosenbaum, Mathieu, 2013. "Estimating the efficient price from the order flow: A Brownian Cox process approach," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2603-2619.
    19. �lvaro Cartea & Sebastian Jaimungal, 2013. "Modelling Asset Prices for Algorithmic and High-Frequency Trading," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(6), pages 512-547, December.
    20. Pietro Fodra & Mauricio Labadie, 2012. "High-frequency market-making with inventory constraints and directional bets," Working Papers hal-00675925, HAL.
    21. Monique Jeanblanc‐Picqué, 1993. "Impulse Control Method and Exchange Rate," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 161-177, April.
    22. Matteo Basei, 2019. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 355-383, June.
    23. K. Giesecke & H. Kakavand & M. Mousavi, 2011. "Exact Simulation of Point Processes with Stochastic Intensities," Operations Research, INFORMS, vol. 59(5), pages 1233-1245, October.
    24. Erhan Bayraktar & Mike Ludkovski, 2009. "Optimal Trade Execution in Illiquid Markets," Papers 0902.2516, arXiv.org.
    25. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, January.
    26. Cadenillas, Abel & Zapatero, Fernando, 1999. "Optimal Central Bank Intervention in the Foreign Exchange Market," Journal of Economic Theory, Elsevier, vol. 87(1), pages 218-242, July.
    27. Christian Yann Robert & Sylvain Delattre & Mathieu Rosenbaum, 2013. "Estimating the efficient price from the order flow: A Brownian Cox process approach," Post-Print hal-02006747, HAL.
    28. Mundaca, Gabriela & Oksendal, Bernt, 1998. "Optimal stochastic intervention control with application to the exchange rate," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 225-243, March.
    29. Jerome F. Eastham & Kevin J. Hastings, 1988. "Optimal Impulse Control of Portfolios," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 588-605, November.
    30. Olivier Gu'eant, 2016. "Optimal market making," Papers 1605.01862, arXiv.org, revised May 2017.
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