= Solution
{c}
For the <forward-start call option>, the strike is fixed by the intermediate <stock> price. The four path payoffs are
$$
\begin{array}{c|c|c|c}
\text{path}&S_1&S_2&(S_2-S_1)^+\\\hline
uu&12&16&4\\
ud&12&8&0\\
du&6&8&2\\
dd&6&4&0
\end{array}
$$
The same final price eight has different payoffs on the two middle paths, so one must retain the history at expiry. At time one, <risk-neutral pricing> gives $C_1^u=(3/10)(2\cdot4)=12/5$ and $C_1^d=(3/10)(2\cdot2)=6/5$. Thus
$$
\boxed{C_0=\frac3{10}\left(2\cdot\frac{12}{5}+\frac65\right)=\frac95=1.8.}
$$
Equivalently, its value at the strike-fixing date is $S_1/5$ by homogeneity, so the <martingale> property of discounted <stock> prices gives the initial value $S_0/5$.
\Image[/past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2015-binomial-exercise-and-forward-start.png]
{title=Two-step American-put exercise and forward-start call payoffs}
{description=The American put is exercised at the down node after one step. The forward-start call has different payoffs on the two paths ending at stock price eight, because their strikes were fixed at different intermediate prices.}
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