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Risk of a position built in stages

Market makingHard

A mid price follows a Brownian motion $W_{t}$ with $W_{0} = 0$ and variance $1$ per unit of time. You buy $2$ lots at time $0$ and $1$ more lot at time $3$ , then sell everything at time $5$ . Your P&L is $2 W_{5} + (W_{5} - W_{3})$ . What is its variance? Give a whole number.

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Hint

Rewrite the P&L as coefficients on independent increments.

Solution

On $[0, 3]$ you hold $2$ lots, which adds $2^{2} \times 3 = 12$ to the variance.
On $[3, 5]$ you hold $3$ lots, which adds $3^{2} \times 2 = 18$ .
Independent increments add: $12 + 18 = 30$ .

Answer: $30$

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