Optimal spread when fill rate falls off linearly

Question

You quote a symmetric market at a half-spread $\delta$ around fair value. This time model the fill intensity as falling off linearly: $\lambda(\delta) = A(1 - k\delta)$ for $0 \le \delta \le 1/k$ , and zero beyond, so flow dries up completely once you quote wider than $1/k$ . Each fill earns you the half-spread $\delta$ .

What half-spread maximizes expected profit per unit time? Then redo it when each fill carries an adverse-selection cost $c$ .

Accepted Answer

No adverse selection. Profit rate is money-per-fill times fills-per-unit-time: P(\delta) = \delta \cdot A(1 - k\delta) = A(\delta - k\delta^2). Differentiate and set to zero: P'(\delta) = A(1 - 2k\delta) = 0 \implies \boxed{\delta^ = \frac{1}{2k}.} This sits exactly halfway to the choke point 1/k, the classic result for linear demand: quote at the midpoint between "free" and "no flow at all." As with the exponential model, the optimum depends only on the price-sensitivity k, not the flow level A. With adverse selection. If each fill also costs an expected c (informed flow moves value against …

Optimal spread when fill rate falls off linearly

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