$$b$.$ Find the hedge ratio, construct a hedged portfolio using stocks and call and prove that this is a hedged portfolio. $(10$ points$)$

h$=$ hedge ratio $=($Cu$-$Cd$)/($Su$-$Sd$)=(3-0)/(33-27)=0\mathrm{.}50$

$(0\mathrm{.}50$S$-$C$)*1000$

Buy $500$ share sell $1000$ calls$$

$500$S$-1000$C

V$=(500*30)-(1000*2\mathrm{.}2642)=12735\mathrm{.}8$

If stock goes up to $33$: Vu$=(500*33)-(1000*3)=13500$

If stock goes down to $27$ Vd$=(500*27)-(1000*0)=13500$

return$=(13500-12735\mathrm{.}8)/12735\mathrm{.}8=0\mathrm{.}0600=6\%$

$6\%=6\%$ risk free rate

is this a hedged portfolio