$($a$)$ Explain two ways in which a bull spread can be created. For both strategies show the profit profile graphically.

$($b$)$ Use put$-$call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.

$($c$)$ Explain two ways in which a short $($or reverse$)$ butterfly spread can be created and plot its payoff pattern at expiration.

$($d$)$ Complete the following table for the short butterfly spread with put options for payoffs at the maturity.

$\backslash$table$[[$Payoffs from Short Butterfly Spread,,,,$],[$Maturity:$,x_1,x_2,x_3,],[$Position $($long$/$short$)$:$,$dots,dots,dots,Total$],[$Number of options:,dots,dots,dots,Pay$-$off$],[dotsdotsdotsdotsdotsdotsdotsdotsdotsdotsdotsdots{S}_T>x_{3}dotsdotsdotsdots{x}_2,$dots,dots,dots,$dots{x}_1,$dots,dots,dots,$dots{S}_T,$dots,dots,dots,dots