Question

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 o = 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: Strike Option Price A r $50 $6.17 0.6274 0.1056 $60 $2.51 0.3430 0.1026 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 to lose or gain?

This is the entire question, there were no tables or any other information given.

Answer 1

Let the hedging portfolio contain A number of stocks and B number of call options with $ 60 at stock price. So, this coupled with original call option (at the money cal option with strike price = current stock price = 50) sold by should have delta as 0 and Gamma as zero.

P = A x S + B x C₆₀ - N x C₅₀

Hence, Delta of P = A x Delta of stock S + B x Delta of C₆₀ - 100 x Delta of C₅₀ = A x 1 + B x 0.3430 - 100 x 0.6247 = 0

Hence, A + 0.343B = 62.47

Gamma of P = A x Gamma of stock S + B x Gamma of C₆₀ - 100 x Gamma of C₅₀ = A x 0 + B x 0.1026 - 100 x 0.1056 = 0

Hence, 0.1026B = 10.56

Hence, B = 10.56 / 0.1026 = 102.92 = 103 (it's a number so it has to be an integral value)

Hence, A = 62.47 - 0.343 x B = 62.47 - 0.343 x 102.92 = 27.17 = 27

Cost of hedge = A x S + B x C₆₀ = 27 x 50 + 103 x 2.51 = $ 1,608.53

------------------------

Since A and B have been rounded off, our final position will not be a perfect hedge.

Overall delta = A x 1 + B x 0.3430 - 100 x 0.6247 = 27 x 1 + 103 x 0.3430 - 100 x 0.6247 = - 0.141

Overall gamma = A x 0 + B x 0.1026 - 100 x 0.1056 = 0 + 103 x 0.1026 - 100 x 0.1056 = 0.0078

------------------------

Amount gained / (Lost) = Delta x change + 1/2 x Gamma x change² = - 0.141 x 5 + 1/2 x 0.0078 x 5² = - $ 0.6075

So, there will be a loss of $ 0.6075