A chooser option is a contract that allows its holder to decide at a later (fixed) time t1 whetherthe option will be a call or a put .The value V(t1 ) of the chooser option at the decision time t1 isthe maximum between the values of the call and put options, that is V(t1 ) = max{ CE (t1 ),PE (t1 )}.

(a)Prove that V( t1 )=CE ( t1 )+max{0,Ker(Tt1) S( t1 )}.

(b)Prove that the price of the chooser option at time 0 must be given by V(0)=CE (0)+PE (0), where CE denotes a European call option with maturity T and strike price K and PE denotes a European put option with maturity t1 and strike price Ker(Tt1) .

(c)Consider a Black-Scholes model with r=0.1, =0.3, S(0)=80, and no dividends. Compute the price of the chooser option, for which the holder can choose at time t1 =0.25 years whether to hold the call or put option with maturity T=0.5 years and strike price K=85.