Question

3. (McDonald Problem 11.19 p. 345) Let S - $40, K = $45, 6 = 30%,r=8%, and 8 = 0%. A dealer has just sold a call option on 100 shares of stock. The option characteristics are as described above and there are 91 days left until expiration a. The dealer wishes to delta-hedge this exposure. i. To hedge, does he need to buy or sell stock? How many shares? ii. How much does he borrow or lend so that the total initial cash flow from the position (including the option) is zero? b. Calculate the overnight profit that will result if the stock price tomorrow is $39. c. Calculate the overnight profit that will result if the stock price tomorrow is $40.50.

Answer 1

(A) Delta hedging also called delta neutral strategy where we earn the profit without using delta, here we use implied volatility and time decay for the option seller.

To use this strategy we sell call and put with same strike or with same delta (weather it OTM, ITM or ATM) or as per the question asked where dealer sell the call option and wants to hedge it

i) suppose he has sold $0.45 delta call option and to hedge this he just buy 45 shares.

ii) his total amount required to invest in call option and buy the stock is Margin money for selling call option and (45 share @ $40)

(B) To calculate overnight profit we need to calculate the option price through the black scholes Model.

The option price as per given input S=$40, K=$45, Volatility is 30%, Risk free rate is 8% and 91 days left in expiry

   is $0.97 and if stock price went down to 39 then option price is $0.70 so the profit is (0.97-0.70)*100 = $27 Net profit

(c) If the stock price tomorrow is $40.5 then the option price is $1.1 and the net profit is (0.97-1.1)*100 = $-13 net profit.