Dark-Pool Perspective of Optimal Market Making

TL;DR

This paper models the optimal market-making strategy for a dark pool, balancing inventory risk, order flow control, and information leakage, by solving a complex impulse-control problem with numerical examples.

Contribution

It introduces a novel stochastic control framework for dark pool market making, incorporating inventory management, order execution uncertainty, and information leakage penalties.

Findings

The value function is the unique viscosity solution of the system of quasi variational inequalities.

Numerical examples demonstrate the model's applicability and include a semi-explicit solution case.

The model effectively balances inventory risk and trading costs in dark pool market making.

Abstract

We consider a finite-horizon market-making problem faced by a dark pool that executes incoming buy and sell orders. The arrival flow of such orders is assumed to be random and, for each transaction, the dark pool earns a per-share commission no greater than the half bid-ask spread. Throughout the entire period, the main concern is inventory risk, which increases as the number of held positions becomes critically small or large. The dark pool can control its inventory by choosing the size of the commission for each transaction, so to encourage, e.g., buy orders instead of sell orders. Furthermore, it can submit lit-pool limit orders, of which execution is uncertain, and market orders, which are expensive. In either case, the dark pool risks an information leakage, which we model via a fixed penalty for trading in the lit pool. We solve a double-obstacle impulse-control problem associated with the optimal management of the inventory, and we show that the value function is the unique viscosity solution of the associated system of quasi variational inequalities. We explore various numerical examples of the proposed model, including one that admits a semi-explicit solution.