$1.$Solve the following by hand: Consider the following options portfolio: You write a January $2012$ expiration call option on IBM with exercise price $$170.$ You also write a January expiration IBM put option with exercise price $$165.$

a$.$ Graph the payoff of this portfolio at option expiration as a function of IBM$$s stock price at that time.

b$.$ What will be the profit$/$loss on this position if IBM is selling at $$167$ on the option expiration date? What if IBM is selling at $$175?$ Refer to Figure $15\mathrm{.}1.$Chapter $15$ Options Markets

$\backslash$table$[[,$Underlying stock price: $166\mathrm{.}76],[$IBM $($IBM$),,$Call,Put,,,,,,,,,,,$],[$Expiration$,$Strike,Last,Volume,Open Interest,Last,Volume,Open Interest,,,,,,,,,,$],[$Sep $2011,160\mathrm{.}00,9\mathrm{.}15,796,1835,2\mathrm{.}62,1271,4298,,,,,,,,,,],[$Oct $2011,160\mathrm{.}00,11\mathrm{.}95,301,2369,5\mathrm{.}35,296,7443,,,,,,,,,,],[$Jan $2012,160\mathrm{.}00,15\mathrm{.}00,101,7890,9\mathrm{.}40,102,4920,,,,,,,,,,],[$Apr $2012,160\mathrm{.}00,17\mathrm{.}35,111,122,13\mathrm{.}30,4,71,,,,,,,,,,],[$Sep $2011,165\mathrm{.}00,5\mathrm{.}80,1805,3542,4\mathrm{.}10,1711,5085,,,,,,,,,,],[$Oct $2011,165\mathrm{.}00,8\mathrm{.}70,329,3866,7\mathrm{.}00,131,2651,,,,,,,,,,],[$Jan $2012,165\mathrm{.}00,11\mathrm{.}70,93,4045,10\mathrm{.}85,303,4466,,,,,,,,,,],[$Apr $2012,165\mathrm{.}00,14\mathrm{.}30,58,781,-,-,54,,,,,,,,,,],[$Sep $2011,170\mathrm{.}00,3\mathrm{.}00,3053,7959,6\mathrm{.}20,2744,12436,,,,,,,,,,],[$Oct $2011,170\mathrm{.}00,5\mathrm{.}86,457,7506,9\mathrm{.}25,148,4584,,,,,,,,,,],[$Jan $2012,170\mathrm{.}00,8\mathrm{.}93,73,6876,13\mathrm{.}00,58,2929,,,,,,,,,,],[$Apr $2012,170\mathrm{.}00,11\mathrm{.}32,36,58,-,-,52,,,,,,,,,,]]$