= Solution
The difference of the terminal <European call option> and <European put option> payoffs is $S_T^1-K$. Replicating that difference by one share and a borrowing position gives conditional <put-call parity>
$$
c_{t_0}-p_{t_0}=S_{t_0}^1-K(1+r)^{t_0-T}.
$$
Thus the call is more valuable exactly when the right side is positive. With a put selected at equality, the <chooser option> payoff is
$$
\boxed{H=(S_T^1-K)^+\mathbf1_{\{S_{t_0}^1>K(1+r)^{t_0-T}\}}
+(K-S_T^1)^+\mathbf1_{\{S_{t_0}^1\leq K(1+r)^{t_0-T}\}}.}
$$
At equality both choices have the same price, even though their eventual payoffs can differ. <Put-call parity> assumes no dividends or other intermediate stock cash flows, as in the stated asset-price model.
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