You are in the Black-Scholes (BS) universe. A chooser option, bought at date t = 0, entitles its owner to wait until date t* (> 0) before deciding whether she possesses a European call, strike Kc and expiry date T_c (> t*) or a European put, strike K_p and expiry date T_p (> t*), both written on the same underlying asset not paying dividends. The choice, once made, cannot be changed. The value of this chooser is given by a complex formula in semi-closed form involving (several times) the bi-variate Gaussian distribution. This is way beyond the scope of the course. However, in the simplified case examined here where K_c = K_p = K and T_c = T_p = T (> t*), the solution is in closed form and involves the usual uni-variate Gaussian only, as in BS. You precisely have to show this. The riskless rate r and the volatility s of the underlying are assumed to be constant.

a) What is the payoff at date t* of this chooser?

b) Can you think of any circumstances under which it would be optimal to choose the call or the put before date t* ?

c) Show that the upper bound for the value of this chooser is the price of the straddle of strike K and expiry date T.

d) Use a well-known relationship to simplify the payoff obtained in a) and recover one or more vanilla payoff(s).

e) Deduce the chooser price at date t = 0 (BS formula is assumed to be known).

f) Check that this price is indeed lower than that of the straddle described in c).