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Order type mixture

Market makingMedium

Orders arrive in $3$ types. With probability $0.5$ an order is small: $1$ unit with an edge of $0.4$ per unit for the market maker. With probability $0.3$ it is medium: $3$ units with an edge of $0.1$ per unit. With probability $0.2$ it is large and informed: $10$ units with an edge of $- 0.12$ per unit. What is the expected profit per order?

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Hint

Convert every type to a profit per order, then weight by probability.

Solution

The profit per order is $0.4$ for small, $0.3$ for medium and $- 1.2$ for large.
The weighted sum is $0.5 \times 0.4 + 0.3 \times 0.3 + 0.2 \times (- 1.2)$ .
The expected profit is $0.2 + 0.09 - 0.24 = 0.05$ .

Answer: $0.05$

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