Question

1. ] An investment bank has the following customers.

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

The investment bank takes the opposite position of the investors. The risk-free interest rate is R_F = 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 bank’s portfolio of gold and its derivatives.

1. Find the current value of P.
2. 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. 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. How can the bank make its portfolio P delta-neutral using spot gold?
5. 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?

Answer 1

P is the current value of portfolio of investment bank, and it can be calculated by formula given as follows:

• V_t(T) = S_t – F₀(T)(1+r)^-(T-t)

We will calculate value of each type of investments for investment bank individually and then add all of them:

A Current value of Investment A:Long forward contract to buy 1000 ounces of gold at HK$10000 per ounce in 3 months

Size of contract = 1000 ounces

Spot price of gold = $9800 /ounce

Forward price = $10000 / ounce  

Risk free interest rate = 5% p.a.

Value = 9800 - 10000/(1+.05)^1/4

Value for investor A= 9800 - 9876.03 = -76.03/ ounce

Value for investment bank = 76.03*1000 = $76030

B Current value of Investment B:Short European put options to sell 500 ounces of gold at HK$10500 per ounce in 6 months

Size of contract = 500 ounces

Spot price of gold = $9800 /ounce

Forward price = $10500 / ounce  

Risk free interest rate = 5% p.a.

Value for investor B= 9800 - 10500/(1+.05)^.5 = 9800 - 10246.95 = -446.85/ounce

Value for investment bank = -(Value for investor B) = 446.85*500 = $223475.4

C Current value of Investment C:Long European put options to sell 2000 Bank of China shares at HK$3.5 in 6 months

Size of contract = 2000 Shares

Spot price of gold = $3/share

Forward price = $3.5/sha9e  

Risk free interest rate = 5% p.a.

Value for investor B= 3.5/(1+.05)^.5 3 = 3.4156-3 = .4156/share

Value for investment bank = -(Value for investor B=C) = -.4156*2000 = -$831.30

Total current value through options= Value through A + Value through B

= $76030 + $223475  

= $ 299505

Value of gold with the investment bank :

As investment bank has agreed to sell gold in investment A and investment B, it must currently have 1500 ounces of gold with it

=1500*9800

= $14700000

Total value of portfolio =

$299505 + $14700000 = $ 14999505

Answer B =

D. Calculation of delta:

if Price of gold per ounce is increased by $1 per ounce :

1. Current value of Investment A:Long forward contract to buy 1000 ounces of gold at HK$10000 per ounce in 3 months

Size of contract = 1000 ounces

Spot price of gold = $9801 /ounce

Forward price = $10000 / ounce  

Risk free interest rate = 5% p.a.

Value = 9801 - 10000/(1+.05)^1/4

Value for investor A= 9801 - 9876.03 = -75.03/ ounce

Value for investment bank = 76.03*1000 = $75030

2. Current value of Investment B:Short European put options to sell 500 ounces of gold at HK$10500 per ounce in 6 months

Size of contract = 500 ounces

Spot price of gold = $9801 /ounce

Forward price = $10500 / ounce  

Risk free interest rate = 5% p.a.

Value for investor B= 9801 - 10500/(1+.05)^.5 = 9800 - 10246.95 = -445.85/ounce

Value for investment bank = -(Value for investor B) = 445.85*500 = $222925