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CHAPTER 3 Futures Prices

CHAPTER 3 Futures Prices. In this chapter, we discuss how futures contracts are priced. This chapter is organized into the following sections: Reading Futures Prices The Basis and Spreads Models of Futures Prices Futures Prices and Expectations Future Prices and Risk Aversion

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CHAPTER 3 Futures Prices

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  1. CHAPTER 3Futures Prices • In this chapter, we discuss how futures contracts are priced. This chapter is organized into the following sections: • Reading Futures Prices • The Basis and Spreads • Models of Futures Prices • Futures Prices and Expectations • Future Prices and Risk Aversion • Characteristics of Futures Prices Chapter 3

  2. Reading Futures Prices • TERMINOLOGY • To understand how to read the Wall Street Journal futures price quotations, we need to first understand some terminology. • Spot Price • Spot price is the price of a good for immediate delivery. • Nearby Contract • Nearby contracts are the next contract to mature. • Distant Contract • Distant contracts are contracts that mature sometime after the nearby contracts. Chapter 3

  3. Reading Futures Prices • TERMINOLOGY • Settlement Price • Settlement price is the price that contracts are traded at the end of the trading day. • Trading Session Settlement Price • New term used to reflect round-the-clock trading. • Open Interest • Open interest is the number of futures contracts for which delivery is currently obligated. Chapter 3

  4. Reading Futures Prices • Insert figure 3.1 Here Chapter 3

  5. How Trading Affects Open Interest The last column in Figure 3.1 shows the open interest or total number of contracts outstanding for each maturity month. Assume that today, Dec 1997, widget contract has just been listed for trading, but that the contract has not traded yet. Table 3.1 shows how trading affects open interest at different times (t). Chapter 3

  6. Open Interest &Trading Volume Patterns • Insert Figure 3.2 Here • Insert Figure 3.3 Here Chapter 3

  7. The Basis The Basis The basis is the difference between the current cash price of a commodity and the futures price for the same commodity. • S0 = current spot price • F0,t = current futures price for delivery of the product at time t. • The basis can be positive or negative at any given time. • Normal MarketPrice for more distant futures are higher than for nearby futures. • Inverted MarketDistant futures prices are lower than the price for contracts nearer to expiration. Chapter 3

  8. The Basis • Example: if the current price of gold in the cash market is $353.70 (July 11) and a futures contract with delivery in December is $364.20. How much is the basis? Chapter 3

  9. The Basis • Insert Figure 3.4 here • Insert Figure 3.5 here • Convergence • As the time to delivery passes, the futures price will change to approach the spot price. • When the futures contract matures, the futures price and the spot price must be the same. That is, the basis must be equal to zero, except for minor discrepancies due to transportation and other transactions costs. • The relatively low variability of the basis is very important for hedging. Chapter 3

  10. Spreads Spread A spread is the difference in price between two futures contracts on the same commodity for two different maturity dates: • Where • F0,t = The current futures price for delivery of the product at time t. This might be the price of a futures contract on wheat for delivery in 3 months. • F0,t+k= The current futures price for delivery of the product at time t +k. This might be the price of a futures contract for wheat for delivery in 6 months. • Spread relationships are important to speculators. Chapter 3

  11. Spreads • Suppose that the price of a futures contract on wheat for delivery in 3 months is $3.25 per bushel. • Suppose further that the price of a futures contract on wheat for delivery in 6 months is $3.30/bushel. • What is the spread? • Insert Figure 3.7 Here Chapter 3

  12. Repo Rate • Repo Rate • The repo rate is the finance charges faced by traders. The repo rate is the interest rate on repurchase agreements. • A Repurchase Agreement • An agreement where a person sells securities at one point in time with the understanding that he/she will repurchase the security at a certain price at a later time. • Example: Pawn Shop. Chapter 3

  13. Arbitrage • An Arbitrageur attempts to exploit any discrepancies in price between the futures and cash markets. • An academic arbitrage is a risk-free transaction consisting of purchasing an asset at one price and simultaneously selling it that same asset at a higher price, generating a profit on the difference. • Example: riskless arbitrage scenario for IBM stock trading on the NYSE and Pacific Stock Exchange. • Assumptions: • Perfect futures market • No taxes • No transactions costs • Commodity can be sold short • Price Exchange • Arbitrageur Buys IBM ($105) Pacific Stock E.Arbitrageur Sells IBM $110 NYSERiskless Profit $ 5 Chapter 3

  14. Models of Futures Prices • Cost-of-Carry Model • The common way to value a futures contract is by using the Cost-of-Carry Model. The Cost-of-Carry Model says that the futures price should depend upon two things: • The current spot price. • The cost of carrying or storing the underlying good from now until the futures contract matures. • Assumptions: • There are no transaction costs or margin requirements. • There are no restrictions on short selling. • Investors can borrow and lend at the same rate of interest. • In the next section, we will explore two arbitrage strategies that are associated with the Cost-and-Carry Model: • Cash-and-carry arbitrage • Reserve cash-and-carry arbitrage Chapter 3

  15. 0 1 1. Borrow money2. Sell futures contract3. Buy commodity 4. Deliver the commodity against the futures contract5. Recover money & payoff loan Cash-and-Carry Arbitrage • A cash-and-carry arbitrage occurs when a trader borrows money, buys the goods today for cash and carries the goods to the expiration of the futures contract. Then, delivers the commodity against a futures contract and pays off the loan. Any profit from this strategy would be an arbitrage profit. Chapter 3

  16. 0 1 1. Sell short the commodity2. Lend money received from short sale3. Buy futures contract 4. Accept delivery from futures contract5. Use commodity received to cover the short sale Reverse Cash-and-Carry Arbitrage • A reverse cash-and-carry arbitrage occurs when a trader sells short a physical asset. The trader purchases a futures contract, which will be used to honor the short sale commitment. Then the trader lends the proceeds at an established rate of interest. In the future, the trader accepts delivery against the futures contract and uses the commodity received to cover the short position. Any profit from this strategy would be an arbitrage profit. Table 3.5 summarizes the cash-and-carry and the reverse cash-and-carry strategies. Chapter 3

  17. Arbitrage Strategies Chapter 3

  18. Cost-of-Carry Model The Cost-of-Carry Model can be expressed as: • Where: • S0 = the current spot price • F0,t = the current futures price for delivery of the product at time t. • C0,t= the percentage cost required to store (or carry) the commodity from today until time t. • The cost of carrying or storing includes: • Storage costs • Insurance costs • Transportation costs • Financing costs • In the following section, we will examine the cost-of-carryrules. Chapter 3

  19. Cost-of-Carry Rule 1 • The futures price must be less than or equal to the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. Chapter 3

  20. 0 1 1. Borrow $4002. Buy 1 oz gold 3. Sell futures contract 4. Deliver gold against futures contract 5. Repay loan Cost-of-Carry Rule 1 Chapter 3

  21. The Cost-of-Carry Rule 2 • The futures price must be equal to or greater than the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. Chapter 3

  22. 0 1 1. Sell short 1 oz. gold2. Lend $420 at 10% interest 3. Buy a futures contract 4. Collect proceeds from loan 5. Accept delivery on futures contract 6. Use gold from futures contract to repay the short sale The Cost-of-Carry Rule 2 Chapter 3

  23. The Cost-of-Carry Rule 3 • Since the futures price must be either less than or equal to the spot price plus the cost of carrying the commodity forward by rule #1. • And the futures price must be greater than or equal to the spot price plus the cost of carrying the commodity forward by rule #2. • The only way that these two rules can reconciled so there is no arbitrage opportunity is by the cost of carry rule #3. • Rule #3: the futures price must be equal to the spot price plus the cost of carrying the commodity forward to the delivery date of the futures contract. If prices were not to conform to cost of carry rule #3, a cash-and carry arbitrage profit could be earned. Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling. Chapter 3

  24. Spreads and The Cost-of-Carry • As we have just seen, there must be a relationship between the futures price and the spot price on the same commodity. • Similarly, there must be a relationship between the futures prices on the same commodity with differing times to maturity. • The following rules address these relationships: • Cost-of-Carry Rule 4 • Cost-of-Carry Rule 5 • Cost-of-Carry Rule 6 Chapter 3

  25. The Cost-of-Carry Rule 4 • The distant futures price must be less than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date. where d > n F0,d = the futures price at t=0 for the distant delivery contract maturing at t=d. Fo,n= the futures price at t=0 for the nearby delivery contract maturing at t=n. Cn,d= the percentage cost of carrying the good from t=n to t=d. If prices were not to conform to cost of carry rule # 4, a cash-and-carry arbitrage profit could be earned. Chapter 3

  26. Spreads and the Cost-of-Carry • Table 3.6 shows that the spread between two futures contracts can not exceed the cost of carrying the good from one delivery date forward to the next, as required by the cost-of-carry rule #4. Chapter 3

  27. 0 1 2 1. Buy futures contract w/exp in 1 yrs. 2. Sell futures contract w/exp in 2 years 3. Contract to borrow $400 from yr 1-2 7. Remove gold from storage8. Deliver gold against 2 yr. futures contract9. Pay back loan 4. Borrow $400 5. Take delivery on 1 yr to exp futures contract. 6. Place the gold in storage for one yr. The Cost-of-Carry Rule 4 Chapter 3

  28. The Cost-of-Carry Rule 5 • The nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date cannot exceed the distant futures price. • Or alternatively, the distant futures price must be greater than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby futures date to the distant futures date. If prices were not to conform to cost of carry rule # 5, a reverse cash-and-carry arbitrage profit could be earned. Chapter 3

  29. The Cost-of-Carry Rule 5 • Table 3.7 illustrates what happens if the nearby futures price is too high relative to the distant futures price. When this is the case, a forward reverse cash-and-carry arbitrage is possible. Chapter 3

  30. 0 1 2 1. Sell futures contract w/exp in 1 yrs. 2. Buy futures contract w/exp in 2 years 3. Contract to lend $400 from yr 1-2 7. Accept delivery on exp 2 yr futures contract 8. Repay 1 oz. borrowed gold. 9. Collect $400 loan 4. Borrow 1 oz. gold 5. Deliver gold on 1 yr to exp futures contract. 6. Invest proceeds from delivery for one yr. The Cost-of-Carry Rule 5 Chapter 3

  31. Cost-of-Carry Rule 6 • Since the distant futures price must be either less than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date by rule #4. • And the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date can not exceed the distant futures price by rule #5. • The only way that rules 4 and 5 can be reconciled so there is no arbitrage opportunity is by cost of carry rule #6. Chapter 3

  32. Cost-of-Carry Rule 6 • The distant futures price must equal the nearby futures price plus the cost of carrying the commodity from the nearby to the distant delivery date. • If prices were not to conform to cost of carry rule #6, a cash-and-carry arbitrage profit or reverse cash-and-carry arbitrage profit could be earned. • Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling. Chapter 3

  33. Implied Repo Rates • If we solve for C0,t in the above equation, and assume that financing costs are the only costs associated with holding an asset, the implied cost of carrying the asset from one time point to another can be estimated. This rate is called the implied repo rate. • The Cost-of-Carry model gives us: Solving for And Chapter 3

  34. Implied Repo Rates • Example: cash price is $3.45 and the futures price is $3.75. The implied repo rate is? • That is, the cost of carrying the asset from today until the expiration of the futures contract is 8.6956%. Chapter 3

  35. The Cost-of-Carry Model in Imperfect Markets • In real markets, no less than four factors complicate the Cost-of-Carry Model: • Direct transactions costs • Unequal borrowing and lending rates • Margin and restrictions on short selling • Limitations to storage Chapter 3

  36. Transaction Costs • Transaction Costs • Traders generally are faced with transaction costs when they trade. In this case, the profit on arbitrage transactions might be reduced or disappear altogether. • Types of Transaction Costs: • Brokerage fees to have their orders executed • A bid ask spread A market maker on the floor of the exchange needs to make a profit. He/She does so by paying one price (the bid price) for a product and selling it for a higher price (the ask price). Chapter 3

  37. Cost-of-Carry Rule 1 with Transaction Costs • Recall that the futures price must be less than or equal to the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. We can modify this rule to account for transaction costs: Where T is the percentage transaction cost. Chapter 3

  38. Cost-of-Carry Rule 1 with Transaction Costs • To show how transaction costs can frustrate an arbitrage consider Table 3.8. Chapter 3

  39. Cost-of-Carry Rule 2with Transaction Costs • Recall from Cost-of-Carry Rule 2 that the futures price must be equal to or greater than the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. We can modify this rule to allow for transaction costs as follows: Chapter 3

  40. Cost-of-Carry Rule 2with Transaction Costs • To show how transaction costs can frustrate an attempt to reserve cash-and-carry arbitrage. Consider Table 3.9. Chapter 3

  41. No-Arbitrage Bounds • Incorporating transaction costs and combining cost-of-carry rules 1 and 2, we have the following. This equation defines the “No Arbitrage Bounds”. That is, as long as the futures price trades within this range, no cash-and-carry or reverse cash-and-carry arbitrage transactions will be profitable. Table 3.10 illustrates this equation. Chapter 3

  42. No-Arbitrage Bounds • In this case, as long as the futures price is between $426.80 and $453.20, arbitrage transactions will not be profitable. Chapter 3

  43. $453.20 Futures Price $426.80 Time No-Arbitrage Bounds Chapter 3

  44. Differential Transaction Costs • Situations occur where all traders do not have equal transaction costs. • For example, a floor trader, trading on his own behalf would have a lower transaction cost than others. So while he/she might be able to earn an arbitrage profit, others could not. • Such a transaction is called a quasi-arbitrage. Chapter 3

  45. Unequal Borrowing & Lending Rates • Thus far we have assumed that investors can borrow and lend at the same rate of interest. Anyone going to a bank knows that this possibility generally does not exist. • Incorporating differential borrowing and lending rates into the Cost-of-Carry Model gives us: Where: CL = lending rate CB = borrowing rate Chapter 3

  46. Unequal Borrowing & Lending Rates Chapter 3

  47. Restrictions on Short Selling • Thus far we have assumed that arbitrageurs can sell short commodities and have unlimited use of the proceeds. • There are two limitations to this in the real world: • It is difficult to sell some commodities short. • Investors are generally not allowed to use all proceeds from the short sale. • How do limitations on the use of funds from a short sale affect the Cost-of-Carry Model? • We can examine this by editing our transaction cost and differential borrowing Cost-of-Carry Model as follows: Chapter 3

  48. Restrictions on Short Selling • The transaction cost and differential cost of borrowing model is as follows: We modify this by recognizing that you will not get all of the proceeds from the short sale. You will get some portion of the proceeds. Where: ƒ = the proportion of funds received Chapter 3

  49. Restrictions on Short Selling • Table 3.12 illustrates the effect of limitations on the use of short sale proceeds. Chapter 3

  50. $461.44 Futures Price $403.52 Time Restrictions on Short Selling • The effect of the proceed use limitation is to widen the no-arbitrage trading bands. Chapter 3

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