Suppose that an investor can

$($a$)$long$$ one call option; or

$($b$)$short$$ one call option; or

$($c$)$long$$ one put option; or

$($d$)$short$$ one put option.

For these four cases, write $$T

$,$ the profit of this investor at the expiration date$-$T of an

option, as a function of ST

$,$ the spot rate at date$-$T $.$