A stock is selling for $$100$ with a volatility of

$40$ percent. Consider a call option on the stock

with an exercise price of $100$ and an expiration of

one year. The risk$-$free rate is $4\mathrm{.}5$ percent. Let the

call be selling for its Black$-$Scholes$-$Merton

value. You construct a delta$-$hedged position

involving the sale of $10,000$ calls and the pur$-$

chase of an appropriate number of shares. You

can buy and sell shares and calls only in whole

numbers. At the end of the next day, the stock is

at $$99.$ You then adjust your position accordingly

to maintain the delta hedge. The following day

the stock closes at $$102.$ In all cases, use the

spreadsheet BlackScholesMertonBinomial$10$e$.$

xlsm to price the call.

a$.$ Compare the amount of money you end up

with to the amount you would have had if

you had invested the money in a risk$-$free

bond. Explain why the target was or was not

achieved.

b$.$ Now add another option, one on the same

stock with an exercise price of $105$ and the

same expiration. Reconstruct the problem by

delta and gamma hedging. Explain why the

target was or was not achieved.