A forward start option is a contract in which the holder receives at time T$0$ at no extra

cost$,$ an $($European Call type$)$ option with expiry date T$1>$ T$0$ and strike price equal to ST$0$

$.$

Assuming that the underlying price process St

follows a Black$-$Scholes model and the risk

free rate is r$,$ find the price F St of such an option at any times t in $[0,$ T$1]$