The Black$-$Scholes formula for a call option with strike price $x$ and time to expiry $T$ is

$c_0(x,T)=S_0[d_1]-$exp$[-yT]x[d_2].$

$($a$)$ Show that its derivatives satisfy:

$S_0{}^{\text{'}}[d_1]exp[yT]=x{}^{\text{'}}[d_1-\sqrt[\phantom{2}]{T}].$

$($b$)$ Derive the 'Greeks': $,,.$

$($c$)$ Find theta $()$ of an option portfolio: long call and short put. $[$Hint: Use the put$-$call parity.$]$

$($d$)$ Find the price of a put option on a non$-$dividend paying stock if $S_0=100,x=110,T=0\mathrm{.}5,y=0\mathrm{.}05$ and $=0\mathrm{.}35.$ Show each step of the calculation.