  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 and formalized mathematically by
Avellaneda and Stoikov. The market is modeled using a reference price $S_t$
following a Brownian motion with standard deviation $\sigma$, 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 approximations based on
a spectral characterization of the optimal quotes.
