= Solution
{c}
At $T_1$, with underlying spot $s$, the <call option on a put option> pays
$$
H(s)=\bigl[p(T_1,s;K_2,T_2)-K_1\bigr]^+.
$$
Let $D=K_2e^{-r(T_2-T_1)}$. The underlying <put option> value is strictly decreasing and convex in $s>0$, with limits $D$ at zero and zero at infinity. If $0<K_1<D$, there is a unique $s_*$ at which that put value equals $K_1$. Hence the compound option payoff is a decreasing convex curve from $D-K_1$ to zero on $[0,s_*]$, followed by a horizontal zero segment. There is an upward kink at $s_*$. If $K_1\geq D$, the payoff is identically zero.
\Image[/past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009-call-on-put.png]
{title=A call on a put has positive payoff only below an exercise threshold}
For $K_1=0$, the holder obtains the underlying put at no cost, so the payoff is its full value at $T_1$. The discounted put price is a <martingale> under the <risk-neutral measure>. Using the <tower property> in the <risk-neutral pricing>,
$$
\boxed{V(t,s)=p(t,s;K_2,T_2),\qquad t<T_1.}
$$
Explicitly, this is $K_2e^{-r(T_2-t)}\Phi(-d_2)-s\Phi(-d_1)$ with $d_1,d_2$ evaluated over $T_2-t$. The intermediate choice time disappears when its exercise cost is zero.
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