How a maker rebate changes your optimal spread

Question

You quote a symmetric market at half-spread $\delta$ with exponential fill intensity $\lambda(\delta) = A\,e^{-k\delta}$ . Each passive fill earns the half-spread $\delta$ , and the exchange also pays a maker rebate $r$ per fill. But the flow is somewhat informed, so each fill also carries an expected adverse-selection cost $c$ .

What half-spread now maximizes expected profit per unit time, and what happens when the rebate is large?

Accepted Answer

Net earnings per fill are the spread plus the rebate minus the toxicity, \delta + r - c. The profit rate is that times the fill rate: P(\delta) = (\delta + r - c)\,A\,e^{-k\delta}. Differentiate: P'(\delta) = A\,e^{-k\delta}\big[1 - k(\delta + r - c)\big] = 0 \implies \boxed{\delta^ = \frac{1}{k} + c - r.} Interpretation. The rebate acts exactly like negative adverse selection. Recall the no-rebate optimum was c + 1/k; the rebate simply subtracts r from it. Getting paid to provide liquidity lets you quote tighter by the full rebate amount, which is precisely why rebate-rich venues show razor-…

How a maker rebate changes your optimal spread

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