Optimal spread with a bell-shaped fill curve

Question

You quote a symmetric market at half-spread $\delta$ around fair value, earning $\delta$ per fill. Suppose the fill intensity decays like a Gaussian in the distance from fair, $\lambda(\delta) = A\,e^{-k\delta^2}$ , so quoting a little wide barely hurts, but quoting far out kills flow very fast.

What half-spread maximizes expected profit per unit time?

Accepted Answer

Profit rate is again edge-per-fill times fill-rate: P(\delta) = \delta \cdot A\,e^{-k\delta^2} = A\,\delta\,e^{-k\delta^2}. Differentiate using the product rule. Since \frac{d}{d\delta}e^{-k\delta^2} = -2k\delta\,e^{-k\delta^2}, P'(\delta) = A\,e^{-k\delta^2}\big(1 - 2k\delta^2\big) = 0. The exponential is never zero, so the bracket sets the optimum: 1 - 2k\delta^2 = 0 \implies \boxed{\delta^ = \frac{1}{\sqrt{2k}}.} Reading the result. Because the penalty for quoting wide is quadratic here rather than linear-in-the-exponent, the optimal spread scales like 1/\sqrt{k} instead of 1/k. Compared w…

Optimal spread with a bell-shaped fill curve

Your answer

More Market Making questions