If the spot gold is at 2000. Suppose the volatility of the return on gold is 30%, and the 90-day risk free rate is 3% per year. Black-Merton-Scholes computations

a) Compute the price of an at-the-money European call option and an at-the money European put option with 90 days to expiration. Also, compute the respective hedge ratios. Use the Black-Merton-Scholes model but do not use a computer program. Show all your specific computations.

b) Repeat a) for the spot prices of 1960, 1980, 2020 and 2040. Assume the same strike (2000).