Liquidation in Limit Order Books With Controlled Intensity

Abstract

We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power‐law and exponential‐decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.

Optimal Placement in a Limit Order Book

“…The idea then is to maximize the profit by playing with the spread between the bid and ask prices, while controlling the inventory risk and the execution risk. See for example, Avelaneda and Stoikov (2008) [8], Bayraktar and Ludkovski (2012) [9], Cartea and Jaimungal (2013) [14], Cartea, Jaimugal, and Ricci (2011) [15], Veraarta (2010) [49], Guilbaud and Pham (2011) [29], Guéant, Lehalla, and Tapia (2012) [28], and Horst, Lehalle and Li (2013) [33]. There are also order scheduling problems especially when orders can be placed in different exchanges with different fee structures.…”

Mini-Flash Crashes, Model Risk, and Optimal Execution

“…Others consider the possibility that their agents' models have the correct form; however, the agents must gradually learn the values of certain unobserved features ( [17], [31], [43], [54], [62], [48]). …”

“…respectively. 31 Conditional onF te−ρ , the RHS of the inequality in (C.21) (and C.22) is deterministic. Since F K,1 (t e ) > 0 (see (C.17)), we finish our proof of Lemma 7.8.…”

Optimal Placement in a Limit Order Book

“…The profit is a reward for the market maker for his or her service to provide the liquidity to the market. For instance, Avellaneda and Stoikov [7], Bayraktar and Ludkovski [8], Cartea and Tutorials in Operations Research, c 2013 INFORMS Jaimungal [13], Cartea et al [14], Veraarta [59], Guilbaud and Pham [31], and Guéant et al [30] study minimizing the inventory risk and balancing the execution risk with consideration of microstructure of LOB. There are also order scheduling problems, especially when orders can be placed in different exchanges with different fee structures.…”