$($Question $1)$ Consider the options written on a tradable underlying asset S that follows the stochastic process $=$ mdt $+$ sd $$ where m and s are constant factors. Assume also that risk$-$free interest rate r is constant and flat. $($a$)$ Derive the current pricing of a down$-$and$-$in European call option with current asset price S$0,$ strike price K$,$ and barrier level L that is above strike price and below current asset price $($K $<$ L $<$ S$0).$ For L $=$ K$,$ evaluate also the current delta D$0=$of this option conditional to asset price S$0.($b$)$ Consider a look$-$back option with payoff at maturity T taken to be $$ fT $=$ max$($Soverall $-$ K$,0),$ for current asset price$$ S$0>$ K $$ where K is a predefined strike price and Soverall is the overall minimum price of the underlying asset during the life of the option. Suppose it has been issued at current time, derive the current pricing of this look$-$back option.