Consider a non$-$dividend$-$paying stock whose price today is $$40.$

$($i$)$ The premium of a European put with a strike of $$30$ and a term of $6$ months is $$3.$

$($ii$)$ The premium of a European put with a strike of $$40$ and a term of $6$ months is $$6.$

$($iii$)$ The premium of a European call with a strike of $$45$ and a $6-$month term is $$3\mathrm{.}5.$

Consider a portfolio that consists of:

$($a$)$ Sell three put options with strike $$40.$

$($b$)$ Buy a put option with strike $$30.$

$($c$)$ Buy two SYNTHETIC $$45$ strike puts that are constructed with appropriate amounts of $$45$ strike calls, stocks, and investing at the risk$-$free rate.