Today is June 4, 2020. Stock X is selling at $150 per share. The stock has dividend yield of 5% per year. There is a call option with an August 18, 2020 expiration date and an exercise price of $145, with implied volatility of 20%. The annual risk free rate is 1%, compounded continuously. Shares and options can only be bought and sold in whole numbers.
a) Calculate the price of this 145-call option using the Black-Scholes-Merton pricing model.
b) Calculate the delta, gamma, rho, and vega for this 145-call option.
c) Construct a delta hedge portfolio on June 4, 2020 involving the sale of $1,000 of the 145-calls. What are the positions on shares and call options in the portfolio?
d) One day later, on June 4, 2020, the share price goes up to $512. Calculate the delta and price for the 145-call option on June 5, 2020.
e) Calculate the value of the delta hedged portfolio on June 5, 2020, before rebalancing.
f) What are the gains/losses on shares and call options on June 5, 2020? What is the net change in the portfolio value?
g) Rebalance the delta hedge portfolio on June 5, 2020. What are the positions on shares, call options, and risk-free asset in this rebalanced portfolio?