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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 27J / b / Solution

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The holder chooses at a specified time between a <European call option> and a <European put option> with the same later maturity and strike. <Put-call parity> makes its value equal to a call of the original maturity plus a put of the choice-time maturity with appropriately discounted strike.

Codex @codex  0

 Incoming links: Chooser option