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Question 4 (10 marks) Suppose the spot price of gold is $1,500 per troy ounce today. The futures price of gold for delivery in 1 year is $1,530 per troy ounce. Assume that the one-year gold futures contract is correctly priced and there are no storage and insurance costs. Also assume that the risk-free rate is compounded annually and you can borrow and lend money at the risk-free rate. a). What is the theoretical parity price of a two-year gold futures contract, assuming a flat yield curve (same interest rates for all maturities)? (2 marks) b). Suppose you short 10 one-year gold futures contracts today and the contract size is 100 troy ounces. The initial margin requited by the clearing house is $10,000 for your 10 futures contracts. After three months, the futures price decreases to $1,525 per troy ounce. Assume that you did not replenish the margin and there was no margin calls over the three months. What is the balance in your margin account in three months? (3 marks) Part c) is not related to Parts a) - b). c). The price of 1-year index futures contract for S&P 500 is 3,000, while the current S&P 500 index value is 2,952. It is also known that the dividend yield on S&P 500 is 1% per year compounded annually. Assume you can borrow and lend money at the riskless rate of 2% per year compound annually. Also, you can buy and short sell the S&P 500 index now. Design an arbitrage strategy and show that the strategy has a zero cost with a sure profit at the end of Year 1. (5 marks)

Answer 1

Flat yield curve indicates no changes to interest rates

F=s(1+r)^n where f is futures rate

Spot price s = 1500

r is interest rate = we need to find

1530=1500(1+r)^1

1+r = 1.02

= 0.02 that is 2%

Theoretical futures price of 2 years

= 1500(1.02)^2 = 1560.6

B) after 3 months futures price is 1525

We shorted it with 10 contracts with 100troy per contract each

Present value of futures contract is = 1530/(1.02)^3/12 = 1522.444

Value of position is 10000+(1525-1522.444)×100×10

= 12556

Margin balance is 12556

Part c)

Futures price is 3000

Spot price = 2952

Risk free rate 2% dividend yield is 1% with annual compounding

F=S+i-d

I is interest and d is dividend yield

= 2952 + 2952(0.02)-2952(0.01)

= 2981.52

Actual futures price is more than theoretical price so futures is overpriced so arbitration exists and arbitration process is long index and short futures contract borrow money

In this strategy we will not loose money because we has hedged our position if index rises then gain is compensated by loss in futures and vice-versa and ultimately we gain 3000-2981.52= 18.48