 Solution

At $T_1$, with underlying spot $s$, the call option on a put option paysLet $D=K_2{e}^{-r(T_2-T_1)}$. 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 $s_{\ast }$ 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 $[0,s_{\ast }]$, followed by a horizontal zero segment. There is an upward kink at $s_{\ast }$. If $K_1\ge D$, the payoff is identically zero.

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,Explicitly, this is $K_2{e}^{-r(T_2-t)}\Phi (-d_2)-s\Phi (-d_1)$ with $d_1,d_2$ evaluated over $T_2-t$. The intermediate choice time disappears when its exercise cost is zero.