A contract is worth 30 now. At the next step it falls by 4 with probability 2/3, and otherwise it rises by x. What value of x makes the contract's value a martingale? Give a whole number.

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A contract is worth $30$ now. At the next step it falls by $4$ with probability $\frac{2}{3}$ , and otherwise it rises by $x$ . What value of $x$ makes the contract's value a martingale? Give a whole number.

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Hint

The expected change over one step must be $0$ .

Solution

The martingale condition is $(\frac{1}{3}) x - (\frac{2}{3}) \times 4 = 0$ .
So the rise is $(\frac{2}{3}) \times \frac{4}{\frac{1}{3}} = 8$ .

Answer: $8$

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