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 FSt of such an option at any times t in $[0,$T$1].$