Static vs Adapted Optimal Execution Strategies in Two Benchmark Trading Models

Abstract

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing with both permanent and temporary impact, minimizing the expected cost of the trade. The second model is a continuous time framework where the objective function is the sum of the expected cost and a value at risk (or expected shortfall) type risk criterion. Optimal adapted solutions are known in both frameworks from the original works of Bertsimas and Lo (1998) and Gatheral and Schied (2011). In this paper we derive the optimal static strategies for both benchmark models and we study quantitatively the improvement in optimality when moving from static strategies to fully adapted ones. We conclude that, in the benchmark models we study, the difference is not relevant, except for extreme unrealistic cases for the model or impact parameters. This indirectly confirms that in the similar framework of Almgren and Chriss (2000) one is fine deriving a static optimal solution, as done by those authors, as opposed to a fully adapted one, since the static solution happens to be tractable and known in closed form.

Mechanics of good trade execution in the framework of linear temporary market impact

“…Proposition 2.2 examines such a reduction, listing its causes. This is novel in the literature and answers the questions raised in Brigo and Di Graziano (2014), Brigo and Piat (2018) and Bellani et al (2018) about the comparison between static and dynamic solutions to the problem of optimal trade execution. The second aspect is the unbiasedness of liquidation errors (Section 2.2).…”

Optimal trading: The importance of being adaptive

“…In this work we investigated trade execution models in which the optimal adaptive strategy differs significantly from the static one. Previous results of Brigo and Piat [6] considered the benchmark models of Bertsimas and Lo with information signal [5] and of Gatheral and Schied [9] after Almgren and Chriss [2]. Under these models the improvement in optimality expected from adaptive strategies was found to be minimal, at least for reasonable values of the model parameters.…”

“…Clearly the class of static strategies is a subset of the class of adaptive strategies, therefore minimizing the cost functional over the class of adaptive strategies is expected to improve the results obtained when minimizing over the static class. In [6] this difference in the costs and in some cases risks was examined for two optimal trading frameworks: the discrete time Bertsimas and Lo model with an information signal and the continuous time Almgren and Chriss model that was studied by Gatheral and Schied in [9]. In both frameworks, the difference between the transaction costs resulting from the optimal adaptive strategies and the corresponding optimal static strategies were negligible, except in cases where one took unrealistic parameter values for either the asset dynamics or the market impact function.…”

“…In both frameworks, the difference between the transaction costs resulting from the optimal adaptive strategies and the corresponding optimal static strategies were negligible, except in cases where one took unrealistic parameter values for either the asset dynamics or the market impact function. One of the main questions which was left open in [6], was whether there is any optimal trading framework in which the difference between the costs of adaptive vs static strategies will be considerable in a realistic setting. The main goal of this paper is to point out one such trading framework.…”

Optimal Liquidation with Signals: the General Propagator Case

“…From the modelling point of view, incorporating signals into execution problems translates into taking into consideration a non-martingale price process, which changes the problem significantly. The resulting optimal strategies in this setting are often random and in particular signal-adaptive, in contrast to deterministic strategies, which are typically obtained in the martingale price case [13,10]. Results on optimal trading with signals but without a transient price impact component (i.e.…”