A stock trades at $$20.$ Its annual volatility is $18\%.$ The risk$-$free rate is $3\%.$

$($a$)$ Calculate the price of a European call option and put option with strike $K=20$ and $T$ equal to $4$ months.

$($b$)$ Calculate the $$ of a portfolio consisting of $2$ long calls and $1$ short put from the previous part. How many shares of the stock would you short in order to build a new portfolio that is delta$-$neutral?

$($c$)$ What is the

$u$ of one of these calls? of one of the puts?

$($d$)$ Calculate the $$ and the

$u$ of a straddle built from one of the calls and one of the puts. This example illustrates why a straddle built from at$-$the$-$money options is close to being deltaneutral and is a bet on volatility.