Crisafi, MA; Macrina, A; (2017) Inventory Management in Customised Liquidity Pools. Applied & Interdisciplinary Mathematics , 4 , Article 1281594. 10.1080/23311835.2017.1281594.

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Abstract

We focus on financial firms which offer simultaneous over-the-counter brokerage and dealer services to selected market participants. Such a trading venue is referred to as "customised liquidity pool'' (CLP) in the industry. The dealer activity, whereby two-sided liquidity is offered to a limited pool of clients, shares in common similarities with the market-making problem. The arrival flow of client orders is assumed to be random. The CLP offers a stream of two-way prices to its clients, which are functions of the size traded by the client and the CLP holding in that instant. The main concern is inventory risk, which increases as the number of held positions becomes critically small or large. The CLP can control its inventory by choosing the size and the skew of its spread, so to encourage, e.g., buy orders instead of sell orders. Furthermore, it can submit limit orders to standard exchanges, of which execution is uncertain, and market orders, which are expensive. In either case, the CLP risks an information leakage, which is discouraged via a fixed penalty for trading in the standard exchange. We numerically 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 with increasing complexity. We observe that the CLP skews its spread before resorting to limit orders. Ultimately it will cross the spread in the standard exchange and submit aggressive orders. We learn that it is optimal to post limit orders deeper in the book when the inventory is relatively small and progressively move towards the best price as the inventory increases---the hedging strategy crucially depends on the width of the spread in the standard exchange. The CLP will adjust its pricing-hedging strategy according to its P&L target, a feature we analyse by considering various degrees of the CLP's risk appetite.

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