A chooser option allows the owner to decide on \(T_{1}\) whether to own a European-style \(K\)-strike call or put option with expiry \(T_{2}>T_{1}\).

(a) At \(T_{1}\), the payoff of the chooser option is

\[\max \left(C\left(T_{1}, T_{2}, Kight), P\left(T_{1}, T_{2}, Kight)ight)\]

Provide a formula for the chooser option by using the put-call parity at \(T_{1}\).

(b) Show that the chooser option becomes a straddle when \(T_{1}=T_{2}\).

(c) Let \(A(0)=100, r=4 \%, \sigma=12 \%\), and price a 6 -month final expiry \(\left(T_{2}=0.5ight) \operatorname{ATMF}\left(K=F_{A}(0,0.5)ight)\) chooser option where the option holder chooses the option type in three months \(\left(T_{1}=0.25ight)\).