When to Cross the Spread? Trading in Two-Sided Limit Order Books

In this paper the problem of optimal trading in illiquid markets is addressed when the deviations from a given stochastic target function describing, for instance, external aggregate client flow are penalized. Using techniques of singular stochastic control, we extend the results of [F. Naujokat and N. Westray, Math. Financ. Econ., 4 (2011), pp. 299--335] to a two-sided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. In addition to using market orders, the trader can also submit orders to a dark pool. 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 characterization of the optimal trading strategy in terms of a nonstandard coupled forward-backward stochastic differential equation (FBSDE). We show that the optimal control can be characterized via buy, sell, and no-trade regions. The new feature is that we now get a nondegenerate no-...

[1]  R. Almgren,et al.  Direct Estimation of Equity Market Impact , 2005 .

[2]  Peter Kratz,et al.  Optimal liquidation in dark pools , 2013 .

[3]  Robert W. Holthausen,et al.  Large-block transactions, the speed of response, and temporary and permanent stock-price effects , 1990 .

[4]  Yuri Kabanov,et al.  Hedging and liquidation under transaction costs in currency markets , 1999, Finance Stochastics.

[5]  Bernt Øksendal,et al.  Singular Stochastic Control and Optimal Stopping with Partial Information of Itô-Lévy Processes , 2012, SIAM J. Control. Optim..

[6]  B. Øksendal,et al.  A maximum principle for optimal control of stochastic systems with delay, with applications to finance. , 2000 .

[7]  A. Kyle Continuous Auctions and Insider Trading , 1985 .

[8]  Steven E. Shreve,et al.  Optimal Execution in a General One-Sided Limit-Order Book , 2011, SIAM J. Financial Math..

[9]  Felix Naujokat Stochastic control in limit order markets , 2011 .

[10]  Robert Almgren,et al.  Optimal execution of portfolio trans-actions , 2000 .

[11]  Alan Kraus,et al.  PRICE IMPACTS OF BLOCK TRADING ON THE NEW YORK STOCK EXCHANGE , 1972 .

[12]  Peter Kratz,et al.  Optimal liquidation in dark pools in discrete and continuous time , 2011 .

[13]  S. Peng,et al.  Reflected solutions of backward SDE's, and related obstacle problems for PDE's , 1997 .

[14]  U. Horst,et al.  On derivatives with illiquid underlying and market manipulation , 2011 .

[15]  Alexander Schied,et al.  Optimal execution strategies in limit order books with general shape functions , 2007, 0708.1756.

[16]  Alexander Schied,et al.  Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets , 2009, Finance Stochastics.

[17]  U. Haussmann,et al.  The stochastic maximum principle for a singular control problem , 1994 .

[18]  H. Waelbroeck,et al.  Optimal Execution of Portfolio Transactions with Short‐Term Alpha , 2013 .

[19]  Jakša Cvitanić,et al.  Reflected forward-backward SDEs and obstacle problems with boundary conditions , 2001 .

[20]  David Mayers,et al.  The effect of large block transactions on security prices: A cross-sectional analysis , 1987 .

[21]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

[22]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[23]  I. Karatzas,et al.  Finite-Fuel Singular Control With Discretionary Stopping , 2000 .

[24]  Nicholas Westray,et al.  Curve following in illiquid markets , 2011 .

[25]  Chester Spatt,et al.  An Empirical Analysis of the Limit Order Book and the Order Flow in the Paris Bourse , 1995 .

[26]  Antje Fruth Optimal order execution with stochastic liquidity , 2011 .

[27]  H. Witsenhausen,et al.  Some solvable stochastic control problemst , 1980 .

[28]  B. Øksendal,et al.  Singular stochastic control and optimal stopping with partial information of jump diffusions , 2010 .

[29]  B. Mezerdi,et al.  A General Stochastic Maximum Principle for Singular Control Problems , 2005 .