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 ForwardsFutures Over-the-counter (OTC)Traded on an exchange Not standardizedStandardized Specific delivery dateRange of delivery dates Settled at end of contractSettled daily Contract is usually deliveredContract is usually closed out before maturity

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 S T, F T = Spot price, Futures price at day T S 0, F 0 = Spot price, Futures price at day T S t, F t = 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: S T – F 0 Short Position: F 0 – S T 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: S T – F 0 Profit on short position: F 0 – S T We want a position that will pay money when rates increase S T 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 F 0 =$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 F t = $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 F 0, agree to “sell” at F t Payoff is F t - F 0 To close a short futures position at time t<T Go long Agree to “buy” at F t, agree to “sell” at F 0 Payoff is F 0 – F t

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 S T <F T Immediately buy asset, short futures contract. At end of day: sell asset for F T Make arbitrage profit of F T -S T Suppose S T >F T Immediately short asset, long futures contract At end of day: buy asset for F T, close short position Make arbitrage profit of S T -F T

20. The futures price at time 0 To determine F 0 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 S 0 (1+r f ) T for the asset r f = rate at which you can borrow money Hence F 0 = S 0 (1+r f ) T

22. Example: Arbitrage Current price of Apache stock: $45 Assume Apache pays no dividends Futures price for Apache stock: Delivery in three months F 0 =45.75 per share r f =5%, which implies S 0 (1+r f ) 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 F 0 =45.25 per share r f =5%, which implies S 0 (1+r f ) 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 r f = 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 = 1000/1.09 10 = 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 F 0 > S 0 (1+r f ) T Now: Take a sort futures position Borrow S 0, buy asset. In future Honor short futures position, sell asset for F 0 Liability has grown to S 0 (1+r f ) T Keep S 0 (1+r f ) T - F 0

31. Summary If F 0 < S 0 (1+r f ) T Now: Take a long futures position Short the asset, deposit proceeds in risk-free account. In future Account has grown to S 0 (1+r f ) T Honor long futures position, buy asset for F 0 Use asset to close out short position Keep S 0 (1+r f ) T - F 0