Optimal quote width

MediumMarket Making & Betting Games~4m

You set a half-spread $s$ (in dollars, $0 \le s \le 4$ ). Wider quotes earn more per fill but scare off flow: the probability a given quote fills is $\left(1 - \dfrac{s}{4}\right)$ , and each fill earns you exactly $s$ . Choose $s$ to maximize your expected capture per quote (fill probability times profit-per-fill). What half-spread $s$ (in dollars) is optimal?

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Expected capture $= (1 - s/4)\cdot s$ - maximize this in $s$ .
It's a downward parabola $s - s^2/4$ ; its peak is at $s = 2$ .

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Optimal quote width

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