The Guilbaud-Pham Model

Classic optimal market-making models like Avellaneda-Stoikov set quotes as a function of inventory and time-to-close, treating the order book as a black box that just fills orders at some rate. The Guilbaud-Pham model extends this by feeding the state of the limit order book itself into the quoting decision — the model tracks the bid-ask spread as it jumps between discrete tick-width states and lets the market maker's optimal action depend on which state the book is currently in.

The added realism is that a market maker in this framework isn't restricted to passively posting limit orders and waiting to be filled — it can also choose to cross the spread with a market order when doing so is worth paying for. When the spread has just widened (a state that tends to mean-revert back to normal), it can be optimal to post more aggressively or even take liquidity, anticipating that the spread will tighten again shortly; when the spread is already tight, patiently resting limit orders on both sides is usually better. The result is a quoting policy that looks less like a smooth function of inventory alone and more like a set of rules keyed to the book's current regime, blending market-making with occasional opportunistic order-taking.

The model sits firmly in the "stochastic control with a Hamilton-Jacobi-Bellman solution" tradition of academic market-making theory — elegant and precise about how spread dynamics should shape quoting, but built on assumptions (a small number of spread states, specific jump intensities) that a real book rarely obeys exactly, which is why practitioners often use it as a conceptual template rather than a plug-in formula.