A butterfly is an option trading strategy that can be used when a trader expects the price of a stock to maintain its current value. A butterfly can be constructed either from put options or call options by

Selling two options at a strike price $K($usually close to the current price of the stock$),$

Buying an option at a higher strike price $K+a,$ and

Buying an option at a lower strike price $K-a.$

Note that the center price $K$ is at the midpoint between the higher and lower prices, $K+$a and $K-a.$

Throughout this problem, you may make all our usual assumptions for mathematical models of financial markets $($in particular, for any security, there is a single price at which we can buy or sell any amount of the security, and that the model allows no arbitrage portfolios$).$

$($a$)$ Suppose that a butterfly is constructed using European call options on a stock $S,$ with expiration date $T.$ Determine the value $V_T$ of the butterfly on the delivery date as a function of the stock price $S_T$ on that day. $($Your expression for $V_T(S_T)$ will also depend on $K$ and $a.)$

$($b$)$ Sketch a graph of the function $V_T(S_T)$ that you found in part $($a$).$

$($c$)$ Now suppose that we construct a butterfly using European put options on the same stock $S,$ also with expiration date $T.$ Determine the value $W_T$ of this butterfly on the delivery date as a function of the stock price $S_T$ on that day. $($Your expression for $W_T(S_T)$ will depend $K$ and $a$ as it did in part $($a$).)$

$2$

$($d$)$ What must be true about the cost to purchase a butterfly constructed from put options vs$.$ the cost to purchase a butterfly constructed from call options?