= Solution
{c}
The terminal stock prices and <European put option> payoffs are
$$
\begin{array}{c|ccc}
\text{state}&uu&ud,du&dd\\\hline
S_2&144&108&81\\
(117-S_2)^+&0&9&36.
\end{array}
$$
Let $q$ be the risk-neutral up probability. <Risk-neutral pricing> and the supplied discounted-price condition give
$$
8=e^{2r}V_0=18q(1-q)+36(1-q)^2,
$$
so $9q^2-27q+14=0$. Its roots are $2/3$ and $7/3$; only the first is a probability. Therefore
$$
\boxed{q=\frac23,\qquad e^r=0.9+0.3q=1.1,\qquad r=\log1.1.}
$$
The current put price is $V_0=8/1.1^2=800/121$ pounds. This interest rate is specific to part (c), and differs from the zero rate in part (b).
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