Analytical Results and Efficient Algorithm for Optimal Portfolio Deleveraging with Market Impact

Abstract

In this paper, we consider an optimal portfolio deleveraging problem, where the objective is to meet specified debt/equity requirements at the minimal execution cost. Permanent and temporary price impact is taken into account. With no restrictions on the relative magnitudes of permanent and temporary price impact, the optimal deleveraging problem reduces to a nonconvex quadratic program with quadratic and box constraints. Analytical results on the optimal deleveraging strategy are obtained. They provide guidance on how we liquidate a portfolio according to endogenous and exogenous factors. A Lagrangian method is proposed to solve the nonconvex quadratic program numerically. By studying the breakpoints of the Lagrangian problem, we obtain conditions under which the Lagrangian method returns an optimal solution of the deleveraging problem. When the Lagrangian algorithm returns a suboptimal approximation, we present upper bounds on the loss in equity caused by using such an approximation.Subject classifications: optimal deleveraging; permanent and temporary price impact; nonconvex quadratic program; Lagrangian method; breakpoint. Area of review: Optimization.

A Partitioning Algorithm for Markov Decision Processes with Applications to Market Microstructure

“…, u i−1 ) and x 0 . Although the pathwise convexity condition is still stronger than Chen et al (2014), in which the objective can be nonconvex pathwise, it is satisfied in our applications. Second, the quadratic cost and linear dynamics in (2) can be generalized to include constant terms and discrete additive random noise.…”

“…The relaxation of convexity is necessary for both applications in this paper. The handling of nonconvex cost subject to an even weaker convexity condition in quadratic programming is highlighted in Chen et al (2014), in which an investor attempts to unwind a portfolio of multiple assets during a single period with a constant trading rate. By comparison, in our application to the optimal order execution, we consider liquidation of a single asset in multiple periods and explicitly model the LOB.…”

“…In the literature on optimal order execution, the LOB dynamics may or may not be explicitly specified. For example, in Bertsimas and Lo (1998), Almgren and Chriss (2001), Almgren (2012), Chen et al (2014), Guo and Zervos (2015), the problem is solved by assuming some price impact functions without directly modeling the LOB. By comparison, Obizhaeva and Wang (2013), Alfonsi et al (2010) solve the optimal execution problem for a one-sided LOB; Tsoukalas et al (2016) study the problem in twosided and block-shaped LOBs of multiple correlated assets; Horst and Naujokat (2014) consider a more general objective, tracking a target function, for a twosided, block-shaped LOB.…”

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

“…In each case Z is taken to be a predictable semimartingale with left limit process Z − and jumps ∆Z = Z − Z − . Models in this category include Ankirchner et al (2016), Brown et al (2010), Chen et al (2014), Gatheral andSchied (2011), Schied (2013), Schied and Schöneborn (2009) Ting et al (2007) in continuous time and Almgren and Chriss (2000), Bertsimas and Lo (1998) in discrete time. Other impact specifications can be found, for example, in Chen et al (2015), Cheridito andSepin (2014), Forsyth (2011), Lorenz and Almgren (2011), Subramanian and Jarrow (2001) and Ting et al (2007).…”

Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions

“…A second strand of literature, Brown et al (2010), Chen et al (2014), Chen et al (2015), identifies the motive to liquidate with a change in market conditions whereby tighter margin requirements lead to lower permitted amount of leverage. The change in market conditions occurs at discrete time points, while the optimal liquidation (deleveraging) is implemented continuously in time.…”