Each round a market maker faces one retail order and one informed order. With a half-spread s, where 0 ≤ s ≤ 4, the retail order trades with probability 1 − s/4 and earns s. The informed order trades only if s < 3, and then costs 3 − s for sure. What is the largest achievable expected profit per round? Give an exact fraction.

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Optimal spread with a toxic order

Market makingHard

Each round a market maker faces one retail order and one informed order. With a half-spread $s$ , where $0 \leq s \leq 4$ , the retail order trades with probability $1 - \frac{s}{4}$ and earns $s$ . The informed order trades only if $s < 3$ , and then costs $3 - s$ for sure. What is the largest achievable expected profit per round? Give an exact fraction.

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Hint

The informed order switches off at one value of $s$ ; check the profit on each side of it.

Solution

For $s < 3$ the profit is $s (1 - \frac{s}{4}) - (3 - s) = 2 s - \frac{s ^{2}}{4} - 3$ , which increases up to $s = 3$ .
For $s \geq 3$ only retail trades, with profit $s (1 - \frac{s}{4})$ , which is decreasing for $s \geq 2$ .
The best choice is $s = 3$ , with profit $3 \times \frac{1}{4} = \frac{3}{4}$ .

Answer: $\frac{3}{4}$

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