A trader creates a bear spread by selling a $3-$month put with a $$50$ strike price for $$4$ and buying a $3-$month put with a $$55$ strike for $$8\mathrm{.}5.$

a$.$ what is the initial investment?

b$.$ what is the total payoff $($excluding initial investment$)$ when the stock price in three months is $($i$)$ $$48,($ii$)$ $$53,$ or $($iii$)$ $$60?$