Exercise $1.$ You just sold an at$-$the$-$money European Call option on a stock that pays no

dividends. The size of this option contract is $100,$ the current stock price is $$50,$ and the

volatility and interest rate parameters are $=25\%$ and $r=5\%.$

You decide to Delta$-$Gamma hedge your portfolio. That is$,$ you plan to buy and sell shares

of the stock and other options in order to have a portfolio that is simultaneously Delta and

Gamma neutral. After looking at the market, you elect to use call options with a strike of

$$60.$

The prices and sensitivites of these options are as follows:

What are the number of shares and call options with a $$60$ strike that you should buy or sell

to hedge the call you sold? What was the cost of your hedge? Is this a perfect Delta$-$Gamma

hedge? If not, what are the overall Delta and Gamma of your portfolio? If in the next $5$

minutes the underlying security appreciates $$5,$ how much do you expect ${?}^1$ to lose or gain?