Dealing with the inventory risk: a solution to the market making problem

Abstract

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll [17] and formalized mathematically by Avellaneda and Stoikov [3]. The market is modeled using a reference price S t following a Brownian motion with standard deviation σ, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S t and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approxi-7 Our results would be mutatis mutandis the same if we added a penalization term −b(|q T |) for the shares remaining at time T . The rationale underlying this point is that price risk prevents the trader from having important exposure to the stock. Hence, qt should naturally mean-revert around 0.

Market Making and Portfolio Liquidation Under Uncertainty

“…The main characteristic of this model is that it does not explicitly consider the limit order book but instead models liquidity statistically, which is an advantage compared to limit order book models when it comes to mathematical tractability. The model in [8,40] has been further complemented by explicitly accounting for inventory constraints [15,33]. More sophisticated models have also been proposed including for example richer dynamics of market orders, impact on the limit order book, adverse selection effects and pre- dictability, to deal with high frequency market making [17].…”

“…Further generalizations include market orders and limit orders at best as well as next to best bid and ask together with stochastic spreads [36]. These models, all dealing with high frequency trading and market making, in the end rely either on first-order approximations or on numerical approximations for the associated partial differential equations, while one particular stylized model manages to provide simple expressions for the optimal quotes of the market maker [33].…”

“…Naturally these problems can be formulated in any of the models suggested in the papers mentioned in the previous section, resulting in a more or less analytically tractable paper and presentation. Based on this we here have deliberately chosen to stay in the more classical settings of [8,40], see also [33], since many of the effects of uncertainty can already be derived in these papers. In particular, using an ansatz similar to [33,34], we are able to essentially explicitly solve the worst-case stochastic control problem formulated below.…”

“…Following [33,34] we also add a bound, denoted by Q > 0, on the inventory the market maker is allowed to hold. In particular, we assume that a market maker with inventory Q will never post a bid quote and symmetrically that a market maker with inventory -Q, i.e.…”

“…the market maker holds a short position of Q shares in the underlying stock, will never post an ask quote. This is a realistic restriction which imposes a risk limit on the inventory and this limit allows us, along the lines of [33,34] to rigorously solve the stochastic control problem to be outlined.…”