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An investment bank has the following customers.

The investment bank takes the opposite position of the investors. The risk-free interest rate is RF = 5% p.a. with continuous compounding. The current spot price of gold is HK$9800 per ounce. The volatility of gold is 12% p.a. The current spot price of a Bank of China share is HK$3.0. The volatility of Bank of China share is 15% p.a. Let P be the investment banks portfolio of gold and its derivatives.

1. (a) Find the current value of P .

2. (b) Find the delta of P. If the price of gold per ounce is increased by $1, what is the

   change of P predicted by delta?

3. (c) Find the gamma of P. If the price of gold per ounce is increased by $1, what is the

   change of delta predicted by gamma?

4. (d) How can the bank make its portfolio P delta-neutral using spot gold?

5. (e) How can the bank make its portfolio P delta-neutral and gamma-neutral using spot gold and a traded call option with delta 0.2 and gamma 0.001?

Investor A Investor B Investor C Long forward contract to buy 1000 ounces of gold at HK$10000 per ounce in 3 months Short European put options to sell 500 ounces of gold at HK$10500 per ounce in 6 months Long European put options to sell 2000 Bank of China shares at HK$3.5 in 6 months Investor A Investor B Investor C Long forward contract to buy 1000 ounces of gold at HK$10000 per ounce in 3 months Short European put options to sell 500 ounces of gold at HK$10500 per ounce in 6 months Long European put options to sell 2000 Bank of China shares at HK$3.5 in 6 months

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