Optimal Execution Strategies in Limit Order Books with General Shape Functions

Abstract

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modeling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

Empirical shape function of limit-order books in the Chinese stock market

“…In contrast, an impatient small trader might submit an small order when the opposite LOB is thin for small ∆'s, since usually he does not have ensuing orders. The optimal trading strategy of a large order also depends on the average LOB shape [7,8], which could be improved if one considers the instant LOB shape function rather than the average.…”

Optimal Placement in a Limit Order Book

“…Later on, the notion of price manipulation is developed as analog of no arbitrage for derivative pricing. Standard references for these models with additive and possible nonlinear price impact include Huberman and Stanzl [40], Obizhaeva and Wang [50], Schied and Schöneborn [57], Schied et al [58], Almgren and Lorenz [6], Alfonsi et al [1,2], Gatheral [27], Predoiu et al [55], Schied et al [58], Weiss [62], and works involving the geometric Brownian motion and/or multiplicative price impact include Gatheral and Schied [28], Forsyth et al [23], and Guo and Zervos [33].…”

“…Though optimal execution and optimal placement are two different problems, the former is sometimes studied with the incorporation of certain aspects of the latter, especially when the LOB is taken into consideration (see Alfonsi et al [1], Predoiu et al [55]). The latter, on the other hand, could be viewed as the former when the execution risk is removed, as shown by Guo et al [34].…”

Strategic execution in the presence of an uninformed arbitrageur

“…Static policies have also been derived under more complicated models (e.g., Almgren and Chriss, 2000;Huberman and Stanzl, 2005;Obizhaeva and Wang, 2005;Alfonsi et al, 2007b). However, this behavior is in contrast to what is observed amongst institutional traders and trading algorithms that are implemented by practitioners.…”

“…This policy trades equal amounts over time increments within the trading horizon. Further developments have led to optimal execution algorithms for models that incorporate price predictions (Bertsimas and Lo, 1998), bid-ask spreads and resilience (Obizhaeva and Wang, 2005;Alfonsi et al, 2007a), nonlinear price impact models (Almgren, 2003;Alfonsi et al, 2007b), and risk aversion (Subramanian and Jarrow, 2001;Almgren and Chriss, 2000;Dubil, 2002;Huberman and Stanzl, 2005;Engle and Ferstenberg, 2006;Hora, 2006;Almgren and Lorenz, 2006;Lorenz, 2008).…”