Making a Market: the HJB a Naive Quoter Is Ignoring

A market maker who posts a fixed spread around the mid earns that spread on every fill and carries whatever inventory the fills leave behind. We treat the two halves of that sentence as two problems. The naive symmetric quoter is solved in closed form: its expected profit is $2AT\delta e^{-k\delta}$, maximised at a half-spread of $1/k$, and its inventory is a symmetric random walk whose variance grows linearly in the horizon, so that inventory risk outweighs spread-capture risk by a factor $\sigma^2 T / 2\delta^2$ that grows without bound. The optimal quoter is the solution of a Hamilton–Jacobi–Bellman equation which, under an exponential-utility ansatz, collapses to a linear system of ordinary differential equations. We solve that system exactly rather than asymptotically, and two things follow. The optimal quotes are stationary: they settle within a fraction of a unit of time and do not depend on the horizon thereafter. And the closed form in general use is the short-horizon expansion of this solution, wrong by twenty-one per cent in the spread and eighty-seven per cent in the inventory skew at parameters where it is routinely applied. Simulation of all three quoters on the same order flow, scored by certainty equivalent, confirms the ordering and measures the cost: the approximation gives up fifty-six units of certainty equivalent at a horizon of four, and the naive quoter turns negative.