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How long a bracket order stays open

Expected valueMedium

A position's P&L starts at $0$ and moves $+ 1$ or $- 1$ tick each step with equal probability, independently. It closes at $+ 3$ or at $- 5$ , whichever comes first. What is the expected number of steps until it closes? Give a whole number.

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Hint

Apply optional stopping to $S_{t}^{2} - t$ .

Solution

The chance of reaching $+ 3$ first is $\frac{5}{3 + 5} = \frac{5}{8}$ , from optional stopping on $S_{t}$ .
Then $E [S_{τ}^{2}] = (\frac{5}{8}) \times 9 + (\frac{3}{8}) \times 25 = 15$ .
Since $S_{t}^{2} - t$ is a martingale that starts at $0$ , the expected time equals that value: $15$ .

Answer: $15$

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