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An option on the average

Expected valueHard

A stock at $100$ moves up $20%$ or down $10%$ in each of $2$ steps. Cash earns $5%$ per step. An option pays $max (A - 100, 0)$ after step $2$ , where $A$ is the plain average of the prices after step $1$ and step $2$ . What is its no-arbitrage price? Give an exact fraction.

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Hint

Find $q$ , list the $4$ paths, and discount over both steps.

Solution

The up weight is $q = \frac{1.05 - 0.9}{1.2 - 0.9} = \frac{1}{2}$ , so each path has weight $\frac{1}{4}$ .
The paths give averages $\frac{120 + 144}{2} = 132$ , $\frac{120 + 108}{2} = 114$ , $\frac{90 + 108}{2} = 99$ and $\frac{90 + 81}{2} = 85.5$ , so the payoffs are $32$ , $14$ , $0$ and $0$ .
The expected payoff is $\frac{32 + 14}{4} = 11.5$ .
The price is $\frac{11.5}{1.0 5 ^{2}} = \frac{4600}{441}$ .

Answer: $\frac{4600}{441}$

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