A shout option is a European option that allows the holder to “shout” to the writer, at some time τ before maturity, her wish to lock in the current price Sτ of the security. For a call option, the...

A shout option is a European option that allows the holder to “shout” to the writer, at some time τ before maturity, her wish to lock in the current price Sτ of the security. For a call option, the holder’s payoff at maturity, assuming that a shout is made, is

where K is the strike price. (If no shout is made, then the payoff is the usual amount (ST − K) +.) Show that

and use this to find the value of the shout option at time t ≥ τ .