Market Making with Exogenous Competition

We study liquidity provision in the presence of exogenous competition. We consider a `reference market maker' who monitors her inventory and the aggregated inventory of the competing market makers. We assume that the competing market makers use a `rule of thumb' to determine their posted depths, depending linearly on their inventory. By contrast, the reference market maker optimises over her posted depths, and we assume that her fill probability depends on the difference between her posted depths and the competition's depths in an exponential way. For a linear-quadratic goal functional, we show that this model admits an approximate closed-form solution. We illustrate the features of our model and compare against alternative ways of solving the problem either via an Euler scheme or state-of-the-art reinforcement learning techniques.