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When to Cross the Spread: Curve Following with Singular Control

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  • Felix Naujokat
  • Ulrich Horst

Abstract

In this article the problem of curve following in an illiquid market is addressed. Using techniques of singular stochastic control, we extend the results of [NW11] to a twosided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. We first show existence and uniqueness of an optimal control. In a second step, a suitable version of the stochastic maximum principle is derived which yields a characterisation of the optimal trading strategy in terms of a nonstandard coupled FBSDE. We show that the optimal control can be characterised via buy, sell and no-trade regions. The new feature is that we now get a nondegenerate no-trade region, which implies that market orders are only used when the spread is small. This allows to describe precisely when it is optimal to cross the bid ask spread, which is a fundamental problem of algorithmic trading. We also show that the controlled system can be described in terms of a reflected BSDE. As an application, we solve the portfolio liquidation problem with passive orders.

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Bibliographic Info

Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2011-053.

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Length: 31 pages
Date of creation: Aug 2011
Date of revision:
Handle: RePEc:hum:wpaper:sfb649dp2011-053

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Related research

Keywords: Stochastic maximum principle; Convex analysis; Fully coupled forward backward stochastic differential equations; Trading in illiquid markets;

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References

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  1. Lu Lin & Feng Li & Lixing Zhu & Wolfgang Karl Härdle, 2011. "Mean Volatility Regressions," SFB 649 Discussion Papers SFB649DP2011-003, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  2. Mechtenberg, Lydia & Münster, Johannes, 2010. "A strategic mediator who is biased into the same direction as the expert can improve information transmission," Discussion Papers, Research Unit: Market Behavior SP II 2010-19, Social Science Research Center Berlin (WZB).
  3. Markus Reiß & Yves Rozenholc & Charles A. Cuenod, 2011. "Pointwise adaptive estimation for quantile regression," SFB 649 Discussion Papers SFB649DP2011-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  4. Wolfgang Karl Härdle & Vladimir Spokoiny & Weining Wang, 2011. "Local Quantile Regression," SFB 649 Discussion Papers SFB649DP2011-005, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  5. Kartik Anand & Prasanna Gai & Sujit Kapadia & Simon Brennan & Matthew Willison, 2011. "A Network Model of Financial System Resilience," SFB 649 Discussion Papers SFB649DP2011-051, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  6. Heike Belitz & Marius Clemens & Christian von Hirschhausen & Jens Schmidt-Ehmcke & Axel Werwatz & Petra Zloczysti, 2011. "An Indicator for National Systems of Innovation - Methodology and Application to 17 Industrialized Countries," SFB 649 Discussion Papers SFB649DP2011-036, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  7. Rong Liu & Lijian Yang & Wolfgang K. H�rdle, 2013. "Oracally Efficient Two-Step Estimation of Generalized Additive Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 619-631, June.
  8. Alexander Meyer-Gohde, 2011. "Sticky Information and Determinacy," SFB 649 Discussion Papers SFB649DP2011-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  9. Johanna Kappus & Markus Reiß, 2011. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," SFB 649 Discussion Papers SFB649DP2011-027, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  10. Till Strohsal & Enzo Weber, 2011. "Mean-Variance Cointegration and the Expectations Hypothesis," SFB 649 Discussion Papers SFB649DP2011-007, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  11. Wolfgang Karl Härdle & Maria Osipenko, 2011. "Spatial Risk Premium on Weather Derivatives and Hedging Weather Exposure in Electricity," SFB 649 Discussion Papers SFB649DP2011-013, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  12. Gunda-Alexandra Detmers & Dieter Nautz, 2011. "The information content of central bank interest rate projections: Evidence from New Zealand," SFB 649 Discussion Papers SFB649DP2011-032, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  13. Raffaele Fiocco & Carlo Scarpa, 2011. "The Regulation of Interdependent Markets," SFB 649 Discussion Papers SFB649DP2011-046, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  14. Anand, Kartik & Gai, Prasanna & Marsili, Matteo, 2012. "Rollover risk, network structure and systemic financial crises," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1088-1100.
  15. Schied, Alexander & Schoeneborn, Torsten, 2008. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets," MPRA Paper 7105, University Library of Munich, Germany.
  16. Markus Reiß, 2011. "Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise," SFB 649 Discussion Papers SFB649DP2011-028, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  17. Dirk Hofmann & Salmai Qari, 2011. "The Law of Attraction: Bilateral Search and Horizontal Heterogeneity," SFB 649 Discussion Papers SFB649DP2011-017, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  18. Beresford, Alastair R. & Kübler, Dorothea & Preibusch, Sören, 2010. "Unwillingness to Pay for Privacy: A Field Experiment," IZA Discussion Papers 5017, Institute for the Study of Labor (IZA).
  19. Shuzhuan Zheng & Lijian Yang & Wolfgang Karl Härdle, 2011. "A Confidence Corridor for Sparse Longitudinal Data Curves," SFB 649 Discussion Papers SFB649DP2011-002, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  20. Juliane Scheffel, 2011. "Identifying the Effect of Temporal Work Flexibility on Parental Time with Children," SFB 649 Discussion Papers SFB649DP2011-024, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  21. Juliane Scheffel, 2011. "How do Unusual Working Schedules Affect Social Life?," SFB 649 Discussion Papers SFB649DP2011-025, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  22. Esra Akdeniz Duran & Wolfgang Karl Härdle & Maria Osipenko, 2011. "Difference based Ridge and Liu type Estimators in Semiparametric Regression Models," SFB 649 Discussion Papers SFB649DP2011-014, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  23. Raffaele Fiocco & Mario Gilli, 2011. "Bargaining and Collusion in a Regulatory Model," SFB 649 Discussion Papers SFB649DP2011-047, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  24. Esra Akdeniz Duran & Mengmeng Guo & Wolfgang Karl Härdle, 2011. "A Confidence Corridor for Expectile Functions," SFB 649 Discussion Papers SFB649DP2011-004, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  25. Alexander Meyer-Gohde, 2011. "Monetary Policy, Determinacy, and the Natural Rate Hypothesis," SFB 649 Discussion Papers SFB649DP2011-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  26. Wolfgang Karl Härdle & Brenda López Cabrera & Ostap Okhrin & Weining Wang, 2011. "Localising temperature risk," SFB 649 Discussion Papers SFB649DP2011-001, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  27. Sigbert Klinke, 2011. "Developing web-based tools for the teaching of statistics: Our Wikis and the German Wikipedia," SFB 649 Discussion Papers SFB649DP2011-030, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  28. Xiaoliang Liu & Wei Xu & Martin Odening, 2011. "Can crop yield risk be globally diversified?," SFB 649 Discussion Papers SFB649DP2011-018, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  29. Stephan Stahlschmidt & Helmut Tausendteufel & Wolfgang K. Härdle, 2011. "Bayesian Networks and Sex-related Homicides," SFB 649 Discussion Papers SFB649DP2011-045, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
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Citations

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Cited by:
  1. Chen, Ray-Bing & Chen, Ying & Härdle, Wolfgang K., 2014. "TVICA—Time varying independent component analysis and its application to financial data," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 95-109.
  2. Fabian Y. R. P. Bocart & Christian M. Hafner, 2011. "Econometric analysis of volatile art markets," SFB 649 Discussion Papers SFB649DP2011-071, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  3. Hautsch, Nikolaus & Schaumburg, Julia & Schienle, Melanie, 2013. "Financial network systemic risk contributions," CFS Working Paper Series 2013/20, Center for Financial Studies (CFS).
  4. Enno Mammen & Christoph Rothe & Melanie Schienle, 2011. "Semiparametric Estimation with Generated Covariates," SFB 649 Discussion Papers SFB649DP2011-064, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  5. Raffaele Fiocco, 2011. "Competition and regulation in a differentiated good market," SFB 649 Discussion Papers SFB649DP2011-084, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  6. bauwens, Luc & hafner, Christian & pierret, Diane, 2011. "Multivariate volatility modeling of electricity futures," CORE Discussion Papers 2011011, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  7. Nikolaus Hautsch & Ruihong Huang, 2011. "Limit Order Flow, Market Impact and Optimal Order Sizes: Evidence from NASDAQ TotalView-ITCH Data," SFB 649 Discussion Papers SFB649DP2011-056, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  8. Sven Tischer & Lutz Hildebrandt, 2011. "Linking corporate reputation and shareholder value using the publication of reputation rankings," SFB 649 Discussion Papers SFB649DP2011-065, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  9. Gökhan Cebiro˜glu & Ulrich Horst, 2011. "We develop a sequential trade model of Iceberg order execution in a limit order book. The Iceberg-trader has the freedom to expose his trading intentions or (partially) shield the true order size agai," SFB 649 Discussion Papers SFB649DP2011-057, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  10. Gökhan Cebiro˜glu & Ulrich Horst, 2011. "Optimal liquidation in dark pools," SFB 649 Discussion Papers SFB649DP2011-058, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  11. Dorothee Schneider, 2011. "The Labor Share: A Review of Theory and Evidence," SFB 649 Discussion Papers SFB649DP2011-069, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  12. Aurélie Bertrand & Christian M. Hafner, 2011. "On heterogeneous latent class models with applications to the analysis of rating scores," SFB 649 Discussion Papers SFB649DP2011-062, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  13. Gregor Heyne & Michael Kupper & Christoph Mainberger, 2011. "Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators," SFB 649 Discussion Papers SFB649DP2011-067, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  14. Ulrich Horst & Michael Kupper & Andrea Macrina & Christoph Mainberger, 2011. "Continuous Equilibrium under Base Preferences and Attainable Initial Endowments," SFB 649 Discussion Papers SFB649DP2011-082, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  15. Wolfgang Härdle & Maria Osipenko, 2011. "Pricing Chinese rain: a multi-site multi-period equilibrium pricing model for rainfall derivatives," SFB 649 Discussion Papers SFB649DP2011-055, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  16. Patrick Cheridito & Ulrich Horst & Michael Kupper & Traian A. Pirvu, 2011. "Equilibrium Pricing in Incomplete Markets under Translation Invariant Preferences," SFB 649 Discussion Papers SFB649DP2011-083, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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