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 4.5 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 purchase 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 call cases, use the spreadsheet Black Scholes Merton Binomial 10e.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 not achieved?