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A stock at $40$ ends the year at $50$ or $35$ , and cash earns nothing. A real-world forecast gives the up move a $60%$ chance. A claim pays $30$ if the stock ends at $35$ , else nothing. By how much does its no-arbitrage price exceed its real-world expected payoff? Give a whole number.

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Hint

The no-arbitrage price uses the weight $q$ , not the forecast.

Solution

The up weight is $q = \frac{40 - 35}{50 - 35} = \frac{1}{3}$ , so the down weight is $1 - \frac{1}{3} = \frac{2}{3}$ .
The price is $\frac{2}{3} \times 30 = 20$ , and the real expected payoff is $0.4 \times 30 = 12$ .
The excess is $20 - 12 = 8$ .

Answer: $8$

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