4. (25 points total; 5 each) A dealer recently sold a call option with 31 days left until expiration on 100 shares of stock. Option characteristics and market conditions are as described in the table below. Inputs Call Put Stock Price Price 1.9381 0.7981 Exercise Price Delta 0.6488 -0.3512 Volatility 20.000% Gamma 0.1136 0.1136 Risk-free interest rate 3.000% Vega 0.0605 0.0605 Time to Expiration (years) 0.084932 Theta -0.0224 -0.0178 Dividend Yield 0.000% Rho 0.0292 -0.0174 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 decreases to $54.50, resulting in market conditions as described in the table below. Inputs Call Put Stock Price 54.5 Price 1.0768 1.4414 Exercise Price Delta 0.4651 -0.5349 Volatility 20.000% Gamma 0.1272 0.1272 Risk-free interest rate 3.000% Vega 0.0621 0.0621 Time to Expiration (years) 0.082192 Theta -0.0227 -0.0182 Dividend Yield 0.000% Rho 0.0199 -0.0251 55 c. A perfect hedge would have resulted in neither a profit nor a loss on the hedged position. What property of options led to the dealer showing a loss on this delta-hedged position overnight? d. Is there any possible stock price move for which the overnight hedge would have led to a profit? If not, explain why not. If so, indicate qualitatively what types of price moves will lead to a profit, and explain what property of options leads to this profit. e. Given the outcome in (b), if the dealer wishes to continue the delta hedge for another day. how, if at all should the position be adjusted? Specifically: i) How many more shares of stock should she buy or sell? ii) How much more should she borrow or lend? 4. (25 points total; 5 each) A dealer recently sold a call option with 31 days left until expiration on 100 shares of stock. Option characteristics and market conditions are as described in the table below. Inputs Call Put Stock Price Price 1.9381 0.7981 Exercise Price Delta 0.6488 -0.3512 Volatility 20.000% Gamma 0.1136 0.1136 Risk-free interest rate 3.000% Vega 0.0605 0.0605 Time to Expiration (years) 0.084932 Theta -0.0224 -0.0178 Dividend Yield 0.000% Rho 0.0292 -0.0174 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 decreases to $54.50, resulting in market conditions as described in the table below. Inputs Call Put Stock Price 54.5 Price 1.0768 1.4414 Exercise Price Delta 0.4651 -0.5349 Volatility 20.000% Gamma 0.1272 0.1272 Risk-free interest rate 3.000% Vega 0.0621 0.0621 Time to Expiration (years) 0.082192 Theta -0.0227 -0.0182 Dividend Yield 0.000% Rho 0.0199 -0.0251 55 c. A perfect hedge would have resulted in neither a profit nor a loss on the hedged position. What property of options led to the dealer showing a loss on this delta-hedged position overnight? d. Is there any possible stock price move for which the overnight hedge would have led to a profit? If not, explain why not. If so, indicate qualitatively what types of price moves will lead to a profit, and explain what property of options leads to this profit. e. Given the outcome in (b), if the dealer wishes to continue the delta hedge for another day. how, if at all should the position be adjusted? Specifically: i) How many more shares of stock should she buy or sell? ii) How much more should she borrow or lend