Optimal High‐frequency Trading in a Pro Rata Microstructure With Predictive Information

Abstract

We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular controls. We model and discuss the consequences of the two main features of this particular microstructure: first, the limit orders sent by the high frequency trader are only partially executed, and therefore she has no control on the executed quantity. For this purpose, cumulative executed volumes are modelled by compound Poisson processes. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation.The optimal trading problem is studied by stochastic control and dynamic programming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. Next, we examine several situations where we can on one hand simplify the numerical procedure by reducing the number of state variables, and on the other hand focus on specific cases of practical interest. We examine both a market making problem and a best execution problem in the case where the mid-price process is a martingale. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the mid-price is available. Each of the resulting strategies are illustrated by numerical tests.Keywords: Market making, limit order book, pro-rata microstructure, inventory risk, marked point process, stochastic control. 1 IntroductionIn most of modern public security markets, the price formation process, or price discovery, results from competition between several market agents that take part in a public auction. In particular, day trading sessions, which are also called continuous trading phases, consist of continuous double auctions. In these situations, liquidity providers 1 continuously set bid and ask prices for the considered security, and the marketplace publicly displays a (possibly partial) information about these bid and ask prices, along with transactions prices. The action of continuously providing bid and ask quotes during day trading sessions is called market making, and this role was tradionnally performed by specialist firms. However, due to the recent increased availability of electronic trading technologies, as well as regulatory changes, a large range of investors are now able to implement such market making strategies. These strategies are part of the broader category of high frequency trading (HFT) strategies, which are characterized by the fact that they facilitate a larger number of orders being sent to the market per unit of time. HFT takes place in the continuous trading phase, and ...

Portfolio Liquidation in Dark Pools in Continuous Time

“…The trade‐off of execution costs, market risk, and execution risk is also apparent in the models of Bayraktar and Ludkovski (), Guéant et al. (), Guéant and Lehalle (), Guilbaud and Pham (), and Huitema () who consider optimal liquidation with limit orders . In most of these models, the investor chooses a limit price for her orders; the “probability” of execution is then dependent on this price and the left‐over position can be liquidated at fixed costs at the exchange.…”

General Intensity Shapes in Optimal Liquidation

“…Guéant, Lehalle, and Fernandez-Tapia (2012) considered in parallel the specific case of an exponential intensity for a risk-averse agent (see also Laruelle, Lehalle, and Pagès 2011b). More recently, Huitema (2012) considered liquidation involving market orders and limit orders, and Guilbaud and Pham (2015) also propose a liquidation model in a pro-rata microstructure.…”

Optimal trading with alpha predictors

“…Another rather separated stream of literature has studied the use of limit orders both for the liquidation [14,15] and market-making [16]- [20]. The problem of optimally placing market versus limit orders under the possibility of partial execution was studied in [21,22,23].…”

“…One piece missing from most of these articles is the addition of general alpha predictors, as the focus has been mainly marketmaking or liquidation. In [20] some basic predictability was introduced in the price process. Another problem with limit orders is that the optimization problems can be even more intractable that in the case of linear costs (market orders).…”