How wide should you quote to maximize profit?

Question

You quote a symmetric market at a half-spread $\delta$ around your fair value. The rate at which your quotes get filled falls off as you widen: model the fill intensity as $\lambda(\delta) = A\,e^{-k\delta}$ , the further from fair you quote, the exponentially less flow you capture. Each fill earns you the half-spread $\delta$ .

What half-spread maximizes your expected profit per unit time?

Accepted Answer

The trade-off. Widening \delta increases the profit per fill but decreases the number of fills, exponentially. Your expected profit per unit time is the product: P(\delta) = \underbrace{\delta}{	ext{per fill}} 	imes \underbrace{A\,e^{-k\delta}}{	ext{fill rate}} = A\,\delta\,e^{-k\delta}. Maximize: P'(\delta) = A\,e^{-k\delta}\big(1 - k\delta\big) = 0 \implies \boxed{\delta^ = \frac{1}{k}.} The exponential factor is never zero, so the bracket sets the optimum. The best half-spread is 1/k, governed entirely by how sensitive flow is to your pricing (k), and remarkably independent of the overa…

How wide should you quote to maximize profit?

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