(Solution) - A chooser option also known as an as you like it option becomes -(2025 Original AI-Free Solution)

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A chooser option (also known as an as-you-like-it option) becomes a put or call at the discretion of the owner. For example, consider a chooser on the S&R index for which both the call, with value C(St , K, T ? t), and the put, with value P(St , K, T ? t), have a strike price of K. The index pays no dividends. At the choice date, t1, the payoff of the chooser is
max[C(St1, K, T ? t1), P(St1, K, T ? t1)]
a. If the chooser option and the underlying options expire simultaneously, what ordinary option position is this equivalent to?
b. Suppose that the chooser must be exercised at t1 and that the underlying options expire at T. Show that the chooser is equivalent to a call option with strike price K and maturity T plus e??(T?t1) put options with strike price Ke?(r??)(T?t1) and expiration t1.