Optimal spread under power-law flow

Question

You quote a symmetric market at half-spread $\delta$ , and this time model fill intensity as a power law, $\lambda(\delta) = A\,\delta^{-\alpha}$ with $\alpha > 1$ : flow falls off polynomially, with a fat tail of fills even at wide spreads. Each fill earns $\delta - c$ , where $c$ is the adverse-selection cost.

What half-spread maximizes expected profit per unit time, and why does the answer break down if $c = 0$ ?

Accepted Answer

The profit rate is P(\delta) = (\delta - c)\,A\,\delta^{-\alpha} = A\big(\delta^{1-\alpha} - c\,\delta^{-\alpha}\big). Differentiate: P'(\delta) = A\big[(1-\alpha)\delta^{-\alpha} + \alpha c\,\delta^{-\alpha-1}\big] = A\,\delta^{-\alpha-1}\big[(1-\alpha)\delta + \alpha c\big]. The prefactor \delta^{-\alpha-1} is positive, so set the bracket to zero: (1-\alpha)\delta + \alpha c = 0 \implies \boxed{\delta^ = \frac{\alpha}{\alpha - 1}\,c.} Reading the result. With power-law flow the optimal spread is a fixed multiple of the adverse-selection cost, and that multiple \frac{\alpha}{\alpha-1} shrink…

Optimal spread under power-law flow

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