Abstract
We give a complete solution to the problem of minimizing the expected liquidity costs in the presence of a general drift when the underlying market impact model has linear transient price impact with exponential resilience. It turns out that this problem is well-posed only if the drift is absolutely continuous. Optimal strategies often do not exist, and when they do, they depend strongly on the derivative of the drift. Our approach uses elements from singular stochastic control, even though the problem is essentially non-Markovian due to the transience of price impact and the lack in Markovian structure of the underlying price process. As a corollary, we give a complete solution to the minimization of a certain cost-risk criterion in our setting.
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Notes
The requirement that X is bounded is natural from an economic point of view, because the total number of available shares is finite for every stock.
References
Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10, 143–157 (2010)
Alfonsi, A., Schied, A.: Optimal trade execution and absence of price manipulations in limit order book models. SIAM J. Financ. Math. 1, 490–522 (2010)
Alfonsi, A., Schied, A.: Capacitary measures for completely monotone kernels via singular control. SIAM J. Control Optim. 51, 1758–1780 (2013)
Alfonsi, A., Schied, A., Slynko, A.: Order book resilience, price manipulation, and the positive portfolio problem. SIAM J. Financ. Math. 3, 511–533 (2012)
Almgren, R., Chriss, N.: Value under liquidation. Risk 12, 61–63 (1999)
Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–39 (2000)
Bayraktar, E., Ludkovski, M.: Liquidation in limit order books with controlled intensity. Math. Financ. (2011, to appear). doi:10.1111/j.1467-9965.2012.00529.x
Bertsimas, D., Lo, A.: Optimal control of execution costs. J. Financ. Mark. 1, 1–50 (1998)
Blais, M., Protter, P.: An analysis of the supply curve for liquidity risk through book data. Int. J. Theor. Appl. Finance 13, 821–838 (2010)
Bouchard, B., Dang, N.-M., Lehalle, C.-A.: Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financ. Math. 2, 404–438 (2011)
Bouchaud, J.-P., Gefen, Y., Potters, M., Wyart, M.: Fluctuations and response in financial markets: the subtle nature of ‘random’ price changes. Quant. Finance 4, 176–190 (2004)
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential B. Theory of Martingales. North-Holland Mathematical Studies, vol. 72. Herrmann, Paris (1982)
Fruth, A., Schöneborn, T., Urusov, M.: Optimal trade execution and price manipulation in order books with time-varying liquidity. Math. Financ. (2011, to appear). doi:10.1111/mafi.12022
Gatheral, J.: No-dynamic-arbitrage and market impact. Quant. Finance 10, 749–759 (2010)
Gatheral, J., Schied, A.: Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework. Int. J. Theor. Appl. Finance 14, 353–368 (2011)
Gatheral, J., Schied, A., Slynko, A.: Exponential resilience and decay of market impact. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds.) Econophysics of Order-Driven Markets, pp. 225–236. Springer, Berlin (2011)
Gatheral, J., Schied, A., Slynko, A.: Transient linear price impact and Fredholm integral equations. Math. Finance 22, 445–474 (2012)
Guéant, O., Lehalle, C.-A., Fernandez Tapia, J.: Optimal portfolio liquidation with limit orders. SIAM J. Financ. Math. 3, 740–764 (2012)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003)
Kharroubi, I., Pham, H.: Optimal portfolio liquidation with execution cost and risk. SIAM J. Financ. Math. 1, 897–931 (2010)
Klöck, F.: Regularity of market impact models with stochastic price impact. Preprint (2012). http://ssrn.com/abstract=2057610
Lehalle, C.-A., Dang, N.M.: Rigorous post-trade market impact measurement and the price formation process. Trading 1, 108–114 (2010)
Meyer, P.A.: Un cours sur les intégrales stochastiques. In: Séminaire de Probabilités, X (Seconde Partie: Théorie des Intégrales Stochastiques, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975). Lecture Notes in Math., vol. 511, pp. 245–400. Springer, Berlin (1976)
Moro, E., Vicente, J., Moyano, L.G., Gerig, A., Farmer, J.D., Vaglica, G., Lillo, F., Mantegna, R.N.: Market impact and trading profile of hidden orders in stock markets. Phys. Rev. E 80(6), 66–102 (2009)
Obizhaeva, A., Wang, J.: Optimal trading strategy and supply/demand dynamics. J. Financ. Mark. 16, 1–32 (2013)
Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability, vol. 61. Springer, Berlin (2009)
Predoiu, S., Shaikhet, G., Shreve, S.: Optimal execution in a general one-sided limit-order book. SIAM J. Financ. Math. 2, 183–212 (2011)
Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2004)
Schied, A.: Robust strategies for optimal order execution in the Almgren–Chriss framework. Appl. Math. Financ. 20, 264–286 (2013)
Schöneborn, T., Schied, A.: Liquidation in the face of adversity: stealth vs. sunshine trading. Preprint (2009). http://ssrn.com/abstract=1007014
Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics, vol. 140. Am. Math. Soc., Providence (2012)
Acknowledgements
The second author gratefully acknowledges support by Deutsche Forschungsgemeinschaft DFG.
We wish to thank Markus Hess and two anonymous referees for helpful comments on a previous version of the manuscript.
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Lorenz, C., Schied, A. Drift dependence of optimal trade execution strategies under transient price impact. Finance Stoch 17, 743–770 (2013). https://doi.org/10.1007/s00780-013-0211-x
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DOI: https://doi.org/10.1007/s00780-013-0211-x
Keywords
- Optimal trade execution
- Optimal order execution
- Transient price impact
- Singular control
- Verification argument