Optimal Execution Cost for Liquidation Through a Limit Order Market

Abstract

We study the problem of optimally liquidating a large portfolio position in a limit-order market. We allow for both limit and market orders and the optimal solution is a combination of both types of orders. Market orders deplete the order book, making future trades more expensive, whereas limit orders can be entered at more favorable prices but are not guaranteed to be filled. We model the bid-ask spread with resilience by a jump process, and the market-order arrival process as a controlled Poisson process. The objective is to minimize the execution cost of the strategy. We formulate the problem as a mixed stochastic continuous control and impulse problem for which the value function is shown to be the unique viscosity solution of the associated variational inequalities. We conclude with a calibration of the model on recent market data and a numerical implementation.

Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

“…The first limit follows from the highest order term, w 2 ln w, being multiplied by 1−λr 1−ar > 0 (cf. (17)). On the other hand, the second limit follows from (16):…”

“…For this reason, they have been experiencing a comeback due to a demand for more realistic financial models (e.g. fixed transaction costs and liquidity risk), see for instance [13][14][15][16][17][18][19].…”

A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

“…We are aware of only a few works [46,20] that study zero-sum SDGs with impulse control (along with some related studies on switching controls; e.g., [42,43,7]). This is in spite of the fact that impulse control problems have enjoyed a resurgence (see, e.g., [6,17]) due to a demand for more realistic financial models (e.g., fixed transaction costs and liquidity risk) [34,13,39,18] and their link to backwards stochastic differential equations [33].…”

A Bank Salvage Model by Impulse Stochastic Controls

“…A stochastic impulse control problem for a system perturbed by general jump noise was considered in Bayraktar et al (2013). Finally, impulse optimal controls have been used in concrete financial applications, see Aïd et al (2019); Chevalier et al 2013Chevalier et al , 2016Federico et al (2019); Vath et al (2007).…”

“…We stress that, due to the terminal condition to be imposed, typically a finite horizon stochastic impulse control problem is more difficult to solve than infinite horizon impulse control problems. In fact, an exhaustive literature on stochastic impulse control on an infinite time horizon exists-see, e.g., Bayraktar et al (2013); Belak et al (2017); Chevalier et al (2013); Egami (2008); Guo and Wu (2009); Øksendal and Sulem (2008); Pham (2007); Vath et al (2007); whereas very few results exist for the finite dimensional case-see, e.g., Chevalier et al (2016); Guo and Chen (2013); Tang and Yong (1993).…”