A market maker expects a client order next hour. The posterior for its size is 10 lots with chance 1/5, 20 with chance 2/5, 30 with chance 3/10 and 40 with chance 1/10. Each lot too few costs 3 to cover on screen, and each lot too many costs 1 to unwind. How many lots should be held to minimise the expected cost? Give a whole number.

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Lots to pre-position

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A market maker expects a client order next hour. The posterior for its size is $10$ lots with chance $\frac{1}{5}$ , $20$ with chance $\frac{2}{5}$ , $30$ with chance $\frac{3}{10}$ and $40$ with chance $\frac{1}{10}$ . Each lot too few costs $3$ to cover on screen, and each lot too many costs $1$ to unwind. How many lots should be held to minimise the expected cost? Give a whole number.

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Hint

The best level is a quantile of the posterior, not its mean.

Solution

The target quantile is $\frac{3}{3 + 1} = \frac{3}{4}$ .
The cumulative chances are $\frac{1}{5}$ , then $\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$ , then $\frac{3}{5} + \frac{3}{10} = \frac{9}{10}$ .
The first size to reach $\frac{3}{4}$ is $30$ .

Answer: $30$

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