Futures Contracts

1. Futures Contracts Basics Futures prices Margin Accounts Futures and arbitrage Expected Payoffs Hedging

2. Derivatives • A derivative is a financial instrument whose price depends on the price of another underlying asset. • Major derivative contracts are: • Futures and forward contracts, • Call and put options, • Swaps.

3. Futures Contracts • A futures contract is an agreement to buy or sell an asset at a certain time in the future for a predetermined price F. • F is called the futures price: the price at which you agree to transact in the future. • Both delivery and payment take place on the delivery date. • No money changes hands when the contract is entered. • The current price of any commodity is referred to as the “spot price” (S)

4. Futures Contracts • The party which agrees to buy the asset is said to have taken a “long position” in the futures contract. • The party which agrees to sell the asset is said to have taken a “short position” in the futures contract. • Taking a short position in a futures contract is not the same thing as short-selling an asset.

5. Differences Between Forward and Futures Contracts

6. Notation • S = spot price • F = futures price • T = date the contract expires or matures • On this date the long party delivers the asset • On this date the short party pays F • 0 = today • t = some date after today but before T • ST, FT = Spot price, Futures price at day T • S0, F0 = Spot price, Futures price at day T • St, Ft = Spot price, Futures price at day t

7. Profits of Forward/Futures Contracts • The payoff per unit of a forward/futures contract at the time of delivery: • Long Position: ST – F0 • Short Position: F0 – ST • Futures contracts are usually closed before the contract matures • What is the payoff when contract is closed? • We’ll get to this in a few slides.

8. Bank Balance Sheet • Assets: • Present value: $60M • Modified Duration: 9.43 • YTM = 6% • Liabilities: • Present value: $45M • Modified Duration: 0.96 • YTM=4% • Equity: • Present value: $15M

9. Example: • Suppose that interest rates increase 20bp. How does that affect the equity of the bank? • Value of Assets: • Value of Liabilities: • Value of Equity:

10. Hedging Strategy: • Futures contract on a bond • Profit on long position: ST – F0 • Profit on short position: F0 – ST • We want a position that will pay money when rates increase • ST decreases as interest rates go up • So to hedge, take short position in the futures contract.

11. Hedging • Advantage of hedging with futures contracts • Can loan out to clients the kind of loans they want • Generally long term-fixed rate • Can borrow money on clients terms • Generally short term-variable rate • Hedge out the interest rate risk using futures contracts.

12. Futures prices • Futures are traded on an exchange • Clearinghouse reconciles trades each day and guarantees the transactions • All traders are required to establish a margin account with the clearinghouse.

13. Futures prices • Futures prices are constantly changing and are closely related to spot prices. • Example: • At 10:00am on Monday, you lock in to buy “40,000 pounds of frozen pork bellies, cut and trimmed” • Contract matures on October 31. • Futures price is F0=$1.13 per pound. • You have agreed to pay $45,200 for the 40,000 pounds of pork

14. Futures prices • Example continued • At 2:00pm on Monday, a news story breaks. • “New Pope has commanded all Catholics to stop eating beef and chicken” • Futures price of pork jumps to Ft= $1.678/pound • Enter a short position to sell pork at $1.678/pound • At maturity of the contract, you have locked in a profit (1.678-1.1285)40,000 = 21,980

15. Futures prices • Example Continued: • From the point of view of the clearinghouse, your position is “closed” • You are not required to deliver or take delivery of the pork bellies. • You can withdraw the present value of the profit from your margin account as soon as your position is closed.

16. Closing the futures contract • To close a long futures position at time t<T • Go short • Agree to “buy” at F0, agree to “sell” at Ft • Payoff is Ft - F0 • To close a short futures position at time t<T • Go long • Agree to “buy” at Ft, agree to “sell” at F0 • Payoff is F0 – Ft

17. Short-Selling an Asset • Cash flows when you buy an asset • Time 0: buy asset, pay money (negative flow) • Time 1: sell asset, get money (positive flow) • Cash flows when you short-sell an asset • Time 0: borrow asset, sell, get money (positive flow) • Time 1: buy asset, pay-off liability (negative flow)

18. Example • You hate SPAM, so you short-sell 100 shares of Hormel. • Current price/share = $10 • Borrow shares, sell at market price, get $1000 • One week later, price of Hormel is $8. • Buy back 100 shares at $800. • Return shares to party which loaned them out to you. • Profit: $200.

19. Spot and Futures Prices at Expiration • Arbitrage: Free money • Suppose ST<FT • Immediately buy asset, short futures contract. • At end of day: sell asset for FT • Make arbitrage profit of FT-ST • Suppose ST>FT • Immediately short asset, long futures contract • At end of day: buy asset for FT, close short position • Make arbitrage profit of ST-FT

20. The futures price at time 0 • To determine F0 Use replicating strategy • A long futures position: • Pay out cash in future (say 3 months) • Get asset • Why not do following: • Borrow $$ and buy asset now at current spot price • Pay off loan in three months • Both strategies in three months: • You hold the asset • You pay out a lump sum of cash

21. The futures price at time 0 • Going long the future is identical to borrowing the money and buying the asset now if • Storage costs are not high • The asset pays no dividend or coupons • These are good assumptions for futures contracts on financial assets that pay no dividends • When you borrow $$ to buy the asset now you pay S0(1+rf)T for the asset • rf = rate at which you can borrow money • Hence F0= S0(1+rf)T

22. Example: Arbitrage • Current price of Apache stock: $45 • Assume Apache pays no dividends • Futures price for Apache stock: • Delivery in three months • F0=45.75 per share • rf=5%, which implies S0(1+rf)T=45(1.05)1/4 = 45.55 • How can you take advantage of this arbitrage?

23. Example: Arbitrage • Take short position in futures contract • Borrow $45 at risk-free rate, buy Apache • In three months: • Liability has grown to 45(1.05)1/4 = 45.55 • Honor short futures position, sell Apache for 45.75 • (You already own it) • Left with 45.75-45.55= .20 per share • Free money

24. Example: Arbitrage • Current price of Apache stock: $45 • Assume Apache pays no dividends • Futures price for Apache stock: • Delivery in three months • F0=45.25 per share • rf=5%, which implies S0(1+rf)T=45(1.05)1/4 = 45.55 • How can you take advantage of this arbitrage?

25. Example: Arbitrage • Take long position in futures contract • Short Apache, get $45 • Deposit money in risk-free account • In three months: • Money has grown to 45(1.05)1/4 = 45.55 • Honor long futures position, buy Apache for 45.25 • Use to close out short position • Left with 45.55-45.25= 0.30 per share • Free money

26. Arbitrage Example • Assume one interest rate rf = 9%. • This is the rate at which you can both borrow and lend. • Zero-coupon bond: • Matures in 10 years. • YTM: 9% • FV=1000 • Price = 1000/1.0910 = 422.41

27. Arbitrage Example • Futures contract • Delivery of 1 zero-coupon bond in six months • The futures price must be 422.41(1.09)1/2 = 441.01 or else there is an “arbitrage opportunity”

28. Arbitrage Example • Suppose futures price is 450 > 441.01 • Attack the arbitrage • Short the futures contract • Borrow $422.41 and buy the bond. • In six months • Honor short futures position – sell bond for 450 • Liability has grown to 422.41 ´ (1.09)1/2 = 441.01 • Keep excess: 450.00-441.01 = 8.99

29. Arbitrage Example • Suppose futures price is 420 < 441.01 • Attack the arbitrage • Long the futures contract • Short the bond and loan out proceeds at risk-free rate. • Depost 422.41 in risk-free account • In six months • Honor long futures position – buy bond for 420 • Account has grown to 422.41 ´ (1.09)1/2 = 441.01 • Use bond to close out short bond position • Keep excess: 441.01-420 = 21.01

30. Summary • If F0 > S0(1+rf)T • Now: • Take a sort futures position • Borrow S0, buy asset. • In future • Honor short futures position, sell asset for F0 • Liability has grown to S0(1+rf)T • Keep S0(1+rf)T - F0

31. Summary • If F0 < S0(1+rf)T • Now: • Take a long futures position • Short the asset, deposit proceeds in risk-free account. • In future • Account has grown to S0(1+rf)T • Honor long futures position, buy asset for F0 • Use asset to close out short position • Keep S0(1+rf)T - F0