 Solution

Put $K^{\prime }=K(1+r{)}^{t_0-T}$. At the choice time,Buy a European call option of strike $K$ and maturity $T$, together with a European put option of strike $K^{\prime }$ and maturity $t_0$. If the call is selected, the short-maturity put expires worthless. If the put is selected, sell the call and combine its value with the short-maturity put's cash payment; put-call parity makes this exactly enough to buy the maturity-$T$ put. This self-financing replication provesIt requires tradability of the priced options at the choice time, but not completeness of every other market claim. The identity also covers $t_0=0$ and $t_0=T$.