1. 0 DERIVATIVES WORKBOOK By Ramon Rabinovitch

2. 1 DERIVATIVES ARE CONTRACTS Two parties Agreement Underlying security

3. 2 DERIVATIVES FORWARDS FUTURES OPTIONS SWAPS

4. 3 A FORWARDIS A BILATERAL AGREEMENT IN WHICH ONE PARTY COMMITS TO BUY AND THE OTHER PARTY COMMITS TO SELL A SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR A PREDETERMINED PRICE ON A SPECIFIC DATE IN THE FUTURE.

5. 4 A FUTURES IS NOTHING MORE THAN A STANDARDIZED FORWARD TRADED ON AN ORGANIZED EXCHANGE. STANDARDIZATION THE COMMODITY TYPE AND QUALITY THE QUANTITY PRICE QUOTES DELIVERY DATES DELIVERY PROCEDURES

6. 5 AN OPTION IS A BILATERAL AGREEMENT IN WHICH ONE PARTY HAS THE RIGHT, BUT NOT THE OBLIGATION, TO BUY OR SELL A SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR A PREDETERMINED PRICE BEFORE OR ON A SPECIFIC DATE IN THE FUTURE. THE OTHER PARTY HAS THE OBLIGATION TO DO WHAT THE FIRST PARTY WISHES TO DO. THE FIRST PARTY, HOWEVER, MAY CHOOSE NOT TO EXERCISE ITS RIGHT AND LET THE OPTION EXPIRE WORTHLESS.

7. 6 A SWAP IS A BILATERAL AGREEMENT IN WHICH THE TWO PARTIES COMMIT TO EXCHANGE A SERIES OF CASH FLOWS. THE CASH FLOWS ARE BASED ON AN AGREED UPON PRINCIPAL AMOUNT. NORMALLY, ONLY THE NET FLOW EXCHANGES HANDS.

8. 7 WHY TRADE DERIVATIVES? THE FUNDAMENTAL REASON FOR TRADING FORWARDS AND FUTURES IS : PRICE RISK or VOLATILITY

9. 8 PRICE RISK IS THE VOLATILITY ASSOCIATED WITH THE COMMODITY’S PRICE IN THE CASH MARKET REMEMBER THAT THE CASH MARKET IS WHERE FIRMS DO THEIR BUSINESS. I.E., BUY AND SELL THE COMMODITY. ZERO PRICE VOLATILITY NO DERIVATIVES!!!!

10. 9 0 Pr S0S0 t time StSt PRICE RISK At time zero the commodity’s price at time t is not known.

11. 10 0 Pr S0S0 t time StSt PRICE RISK The larger the volatility, the more need for derivatives

12. 11

13. 12

14. 13

15. 14

16. 15 THE ECONOMIC PURPOSES OF DERIVATIVE MARKETS HEDGING PRICE DISCOVERY SAVING HEDGING IS THE ACTIVITY OF MANAGING PRICE RISK EXPOSURE PRICE DISCOVERY IS THE REVEALING OF INFORMSTION ABOUT THE FUTURE CASH MARKET PRICE FOR A PRODUCT. SAVING IS THE COST SAVING ASSOCIATED WITH SWAPING CASH FLOWS

17. 16 ALTHOUGH THE ECONOMIC PURPOSES OF DERIVATIVE MARKETS ARE HEDGING PRICE DISCOVERY SAVING WE WILL SEE THAT SPECULATIVE AND ARBITRAGE ACTIVITIES ARE NOT ONLY BENEFICIAL IN THESE MARKETS, THEY ARE NECESSARY FOR MAINTAINING MARKET EFFICIENCY AND EFFICIENT MARKET PRICES

18. 17 FORWARDS AND FUTURES The CONTRACTS The MARKETS PRICING FUTURES Speculation Arbitrage Hedging

19. 18 Some Financial Economics Principles Arbitrage: A market situation whereby an investor can make a profit with: no equity and no risk. Efficiency: A market is said to be efficient if prices are such that there exist no arbitrage opportunities. Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

20. 19 Valuation: The current market value (price) of any project or investment is the net present value of all the future expected cash flows from the project. One-Price Law: Any two projects whose cash flows are equal in every possible state of the world have the same market value. Domination: Let two projects have equal cash flows in all possible states of the world but one. The project with the higher cash flow in that particular state of the world has a higher current market value and thus, is said to dominate the other project.

21. 20 A proof by contradiction: is a method of proving that an assumption, or a set of assumptions, is incorrect by showing that the implication of the assumptions contradicts these very same assumptions. Risk-Free Asset: is a security of investment whose return carries no risk. Thus, the return on this security is known and guaranteed in advance. Risk-Free Borrowing And Landing: By purchasing the risk-free asset, investors lend their capital and by selling the risk- free asset, investors borrow capita at the risk-free rate.

22. 21 The One-Price Law: There exists only one risk-free rate in an efficient economy.

23. 22 Compounded Interest Any principal amount, P, invested at an annual interest rate, r, compounded annually, for n years would grow to: A n = P(1 + r) n. If compounded Quarterly: A n = P(1 +r/4) 4n. In general, with m compounding periods every year, the periodic rate becomes r/m and nm is the total compounding periods. Thus, P grows to: A n = P(1 +r/m) nm.

24. 23 Monthly compounding becomes: A n = P(1 +r/12) n12 and daily compounding yields: A n = P(1 +r/12) n12. EXAMPLES: n =10 years;r =12%; P = $100 1.Simple compounding yields: A 10 = $100(1+.12) 10 = $310.58 2.Monthly compounding yields: A 10 = $100(1 +.12/12) 120 = $330.03 3.Daily compounding yields: A 10 = $100(1 +.12/365) 3650 = $331.94.

25. 24 In the early 1970s, banks came up with the following economic reasoning: Since the bank has depositors money all the time, this money should be working for the depositor all the time! This idea, of course, leads to the concept of continuous compounding. We want to apply this idea in the formula: Observe that continuous time means that the number of compounding periods, m, increases without limit, while the periodic interest rate, r/m, becomes smaller and smaller.

26. 25 This reasoning implies that in order to impose the concept of continuous time on the above compounding expression, we need to solve: This expression may be rewritten as:

27. 26 EXAMPLE, continued: First, we remind you that the number e is defined as: For example: x e 12 102.59374246 1002.70481382 1,0002.71692393 10,0002.71814592 1,000,0002.71828046 In the limit e = 2.718281828…..

28. 27 Recall that in our example: N = 10 years and r = 12% and P=$100. Thus, P=$100 invested at a 12% annual rate, continuously compounded for ten years will grow to: Continuous compounding yields the highest return to the investor: CompoundingFactor Simple3.105848208 Quarterly3.262037792 Monthly3.300386895 Daily3.319462164 continuously 3.320116923

29. 28 This expression may be rewritten as: Discrete Discounting Clearly, any stream of cash flows may be discounted to the present by discounting every future cash flow for today. P = A n (1 +r/m) -nm. i

30. 29 This expression may be rewritten as: Continuous Discounting

31. 30 EXAMPLE, continued: First, we remind you that the number e is defined as: Recall that in our example: P = $100; n = 10 years and r = 12% Thus, $100 invested at an annual rate of 12%, continuously compounded for ten years will grow to: Therefore, we can write the continuously discounted value of $320.01 is:

32. 31 This expression may be rewritten as: But first, QUESTION: Given P and r, how long it takes to double our money? - “the 72 rule” Ans.: 2P = Pe rt ;t = [ln2]/r t = 69.31/r. r = 10% ==> t = 6.931yrs.

33. 32 PURE ARBITRAGE PROFIT: A PROFIT MADE 1. WITHOUT EQUITY and 2. WITHOUT ANY RISK.

34. 33 Risk-free lending and borrowing Arbitrage: A market situation in which an investor can make a profit with: no equity and no risk. Efficiency: A market is said to be efficient if prices are such that there exist no arbitrage opportunities. Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

35. 34 Risk-free lending and borrowing PURE ARBITRAGE PROFIT: A PROFIT MADE 1.WITHOUT EQUITY INVESTMENT and 2.WITHOUT ANY RISK We will assume that the options market is efficient. This assumption implies that one cannot make arbitrage profits in the options markets

36. 35 Risk-free lending and borrowing Treasury bills: are zero-coupon bonds, or pure discount bonds, issued by the Treasury. A T-bill is a promissory paper which promises its holder the payment of the bond’s Face Value (Par- Value) on a specific future maturity date. The purchase of a T-bill is, therefore, an investment that pays no cash flow between the purchase date and the bill’s maturity. Hence, its current market price is the NPV of the bill’s Face Value: P t = NPV{the T-bill Face-Value} We will only use continuous discounting

37. 36 Risk-free lending and borrowing Risk-Free Asset: is a security whose return is a known constant and it carries no risk. T-bills are risk-free LENDING assets. Investors lend money to the Government by purchasing T-bills (and other Treasury notes and bonds) We will assume that investors also can borrow money at the risk- free rate. I.e., investors may write IOU notes, promising the risk-free rate to their buyers, thereby, raising capital at the risk- free rate.

38. 37 Risk-free lending and borrowing The One-Price Law: There exists only one risk-free rate in an efficient economy. Proof: If two risk-free rates exist in the market concurrently, all investors will try to borrow at the lower rate and simultaneously try to invest at the higher rate for an immediate arbitrage profit. These activities will increase the lower rate and decrease the higher rate until they coincide to one unique risk-free rate.

39. 38 Risk-free lending and borrowing By purchasing the risk-free asset, investors lend capital. By selling the risk-free asset, investors borrow capital. Both activities are at the risk-free rate.

40. 39 We are now ready to calculate the current value of a T-Bill. P t = NPV{the T-bill Face-Value}. Thus: the current time, t, T-bill price, P t, which pays FV upon its maturity on date T, is: P t = [FV]e -r(T-t) Clearly, r is the risk-free rate in the economy.

41. 40 EXAMPLE: Consider a T-bill that promises its holder FV = $1,000 when it matures in 276 days, with a yield-to-maturity of 5%: Inputs for the formula: FV = $1,000 r=.05 T-t= 276/365yrs P t = [FV]e -r(T-t) P t = [$1,000]e -(.05)276/365 P t = $962.90.

42. 41 EXAMPLE:The yield-to -maturity of a bond which sells for $945 and matures in 100 days, promising the FV = $1,000 is: r = ? P t = $945; FV = $1,000; T-t= 100 days. Inputs for the formula: FV = $1,000; P t = $945; T-t = 100/365. Solving P t = [FV]e -r(T-t) for r: r = [365/100]ln[$1,000/$945] r = 10.324%.

43. 42 SHORT SELLING STOCKS An Investor may call a broker and ask to “sell a particular stock short.” This means that the investor does not own shares of the stock, but wishes to sell it anyway. The investor speculates that the stock’s share price will fall and money will be made upon buying the shares back at a lower price. Alas, the investor does not own shares of the stock. The broker will lend the investor shares from the broker’s or a client’s account and sell it in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s request.

44. 43 SHORT SELLING STOCKS Other conditions: The proceeds from the short sale cannot be used by the short seller. Instead, they are deposited in an escrow account in the investor’s name until the investor makes good on the promise to bring the shares back. Moreover, the investor must deposit an additional amount of at least 50% of the short sale’s proceeds in the escrow account. This additional amount guarantees that there is enough capital to buy back the borrowed shares and hand them over back to the broker, in case the shares price increases.

45. 44 SHORT SELLING STOCKS There are more details associated with short selling stocks. For example, if the stock pays dividend, the short seller must pay the dividend to the broker. Moreover, the short seller does not gain interest on the amount deposited in the escrow account, etc. We will use stock short sales in many of strategies associated with options trading. In all of these strategies, we will assume that no cash flow occurs from the time the strategy is opened with the stock short sale until the time the strategy terminates and the stock is repurchased. In terms of cash flows: S t is the cash flow from selling the stock short on date t, and -S T is the cash flow from purchasing the back on date T.

46. 45 THE FORWARDS AND FUTURES MARKETS

47. 46 CASH OR SPOT MARKET THE MARKET FOR IMMEDIATE DELIVERY AND PAYMENT GAS STATION, GROCERY STORE, DEPARTMENT STORE….. SELLER BUYER DELIVERS ACCEPTS COMMODITY NOW ACCEPTSPAYS PAYMENT NOW NOW The SELLER is said to be LONG The BUYER is said to be SHORT

48. 47 A FORWARD MARKET THE MARKET FOR DEFERRED DELIVERY AND DEFFERED PAYMENT. SELLER = SHORT BUYER = LONG THE TWO PARTIES MAKE A CONTRACT THAT DETERMINES THE DELIVERY AND PAYMENT PLACE AND TIME IN THE FUTURE.

49. 48 A FORWARDIS A BILATERAL AGREEMENT IN WHICH ONE PARTY COMMITS TO BUY AND THE OTHER PARTY COMMITS TO SELL A SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR A PREDETERMINED PRICE ON A SPECIFIC DATE IN THE FUTURE.

50. 49 A FUTURES IS NOTHING MORE THAN A STANDARDIZED FORWARD TRADED ON AN ORGANIZED EXCHANGE. STANDARDIZATION THE COMMODITY TYPE AND QUALITY THE QUANTITY PRICE QUOTES DELIVERY DATES DELIVERY PROCEDURES

51. 50 NYMEX Light Crude Oil Futures

52. 51 CBOT Corn Futures

53. 52 CBOT U.S. Treasury Bond Futures

54. 53 CME Standard & Poor’s 500 Stock Index Futures

55. 54 NIKKEI 225 Stock Index Futures

56. 55 The Delivery Sequence for T- Bond Futures Source: Chicago Board of Trade

57. 56 HOW ARE FUTURES CONTRACTS CREATED ? FUTURES CONTRACTS ARE SUGGESTED BY THE FUTURES EXCHANGES THE PROPOPSALS ARE SENT FOR APPROVAL TO THE REGULATORY AUTHORITY: THE FUTURES COMMODITY TRADING COMMISSION. (FCTC)

58. 57 WHY TRADE FUTURES AND NOT FORWARDS? FORWARDS ARE CONTRACTS WITH: Credit risk Operational risk Liquidity risk

59. 58 1.Credit Risk Does the other party have the mean to pay?

60. 59 2.Operational Risk: Will the other party deliver the commodity? Will the other party take delivery? Will the other party pay?

61. 60 3.Liquidity Risk. In case either party wishes to get out of its side of the contract, what are the obstacles? Find another counterparty. It may not be easy to do that. Even if you find someone who is willing to take your side of the contract, the other party may not agree.

62. 61 The exchanges understood that there will exist no efficient markets until the above problems are resolved. So they created the : CLEARINGHOUSE

63. 62 CLEARING MEMBERS NONCLEARING MEMEBRS EXCHANGE CORPORATION CLEARINGHOUSE Futures Commission Merchants CLIENTES THE CLEARINGHOUSE PLACE IN THE MARKET

64. 63 The clearinghouse is a non profit corporation. It gives every trading party an absolute guarantee of the completion of its side of the contract

65. 64 The Clearinghouse guarantee: LONG – will be able to take delivery and pay the agreed upon price. SHORT – will be able to deliver and receive the agreed upon price.

66. 65

67. 66 Outside Customers ABCDE Customer FCM a FCM bFCM c Margins Clearing member 1member 2 ClearinghouseClearinghouse Clearing margins B A } FCM = FUTURES COMMODITY MERCHANT

68. 67 A. BUYER = LONG B. SELLER = SHORT 10 OIL FUTURES 10 OIL FUTURES FOR: $20/ bbl A BUY CH SELL B CLEARINGHOUSE GUARANTEE LONGSHORT BUY 10 JUNE CRUDE $20 SELL 10 JUNE CRUDE THE CH GIVES BOTH A AND B AN ABSOLUTE GUARANTEE OF THEIR SIDE OF THE AGREEMENT. THUS, 1.THERE IS NO CREDIT PROBLEM ! 2.LIQUIDITY PROBLEMS ARE MINIMIZED.

69. 68 BuyerSeller MemberBuying Selling Member firmfloorfloorfirmbroker Trading Ring Buying Orders executed by openSelling flooroutcry by buying and sellingfloor brokerfloor brokers, recorded andbroker confirmsplaced on tickerconfirms purchasesale Member Reports Reports Member firm purchase sale firm Confirms ClearinghouseConfirms Purchase sale 1 ObligationObligation long short Buyer Total open interest 1 contractSeller now long 1contract

70. 69 Seller-longBuyer-short with obligation to pay with obligation for and take deliveryto deliver Member SellingBuyingMember firmfloorfloorfirmbroker Trading Ring SellingBuying floor Orders executed byfloor broker open outcry by buying andbroker confirms selling floor brokers, recordedconfirms sale and placed on tickerpurchase Member ReportsReportsMember firm salepurchasefirm Confirms Clearing HouseConfirms Sale 1 Obligation 1 Obligationpurchase or long or short sold purchased canceling canceling Buyer has buy sellSeller has offset obligation obligationoffset obligationTotal open interestobligation by sale- 0 contractsby purchase-no marketposition

71. 70 Clearing association Member accounts: LongShort FCM (A)250230 FCM (B) * 20 Member FCM (A)Member FCM (B)Customers’ accounts: LongShortLongShort 100 90 0 20 Omnibus accounts: LongShort 150140 Customer 1Customer 2 Customer 3 100 long 90 short 20 short Non-clearing FCM Customer’s accounts: LongShort 150140 Customer 4Customer 5 150 long140 short

72. 71 MARGINS A MARGIN is an amount of money that must be deposited in a margin account in order to open any futures position. It is a “good will” deposit. The clearinghouse maintains a system of margin requirements from all traders, brokers and futures commercial merchants.

73. 72 Most of the time, Initial margins are between 3% to 10% of the position value. Maintenance (or variable) margin is usually around 70% of the initial margin. If, for example, you open a position in 10 CBT treasury bonds futures ($100,000 face value each) at a price of $75,000 each, your initial margin deposit of 5% of $750,000 will stand at $37,500. You will receive a MARGIN CALL when the margin in your margin account will drop to below 75% of this amount or, $26,250.

74. 73 How does your margin changes in the margin account? MARKING TO MARKET Every day, upon the market close, all profits and losses for that day must be SETTLED in cash. The capital in the margin accounts is used in order to settle the accounts, using the SETTLEMENT PRICES

75. 74 A SETTLEMENT PRICE IS the average price of trades during the last several minutes of the trading day. Every day, when the markets close, SETTLEMENT PRICES for the futures of all products and for all months of delivery are set. They are then compared with the previous day settlement prices and the difference must be settled overnight!!!!!!!

76. 75 OPEN A LONG POSITION IN 10 JUNE CRUDE OIL FUTURES AT $18.50/bbl. VALUE: (10)(1,000)($18.50) = $185,000 INITIAL MARGIN = (.03)($185,000) = $5,550 3,750/5,550 =.676 MARGIN CALL ADD $1,800 TO MARGIN ACCOUNT TO BRING IT UP TO $5,550: $ 5,550 DAY 4 $18.97 $189,700 + $6,500 $12,050

77. 76 * A contract: $1M face value of 90-day T-bills. The implied settlement price is 100 - (100 - P)(90/360), where P is the quoted settlement price. ** Without interest earned ** Margin is assumed to be 5% of contract fee. SETTLEMENT PRICES AND MARK-TO-MARKET SETTLEMENTS ON 90-DAY TREASURY BILL FUTURES FOR JUNE 19,1999, SETTLEMENT.

78. 77 JUNE WTI FUTURE 1,000 bbls PER CONTRACT DATE PARTY NUM PRICE PARTY NUM PRICE OI* Th.5.16 A:LONG 10 $20 CH B:SHORT 10 $20 10 5.16 C:LONG 25 $21 CH D:SHORT 25 $21 35 5.16 SETTLE $21 $21 Fr.5.17 E:LONG 10 $22 CH A:SHORT 10 $22 35 5.17 SETTLE $22 $22 Mo.5.20 D:LONG 25 $22.5 CH F:SHORT 25 $22.5 35 5.20 B:LONG 10 $21.5 CH C:SHORT 10 $21.5 25 5.20 SETTLE $21.5 $21.5 Tu.5.21 F:LONG 10 $21 CH E:SHORT 10 $21 15 5.21 SETTLE $21 $21 We.5.22 F:LONG 10 $20 CH C:SHORT 10 $20 5 5.22 SETTLE $20 $20 * OI = Open Interest

79. 78 CLEARINGHOUSE ACCOUNTING A: LONG 10; SHORT 10 : OUT B: SHORT 10; LONG 10 : OUT C: LONG 25; SHORT 10; SHORT 10 C remains LONG 5. D: SHORT 25; LONG 25 : OUT E: LONG 10; SHORT 10 : OUT F: SHORT 25; LONG 10 : LONG 10 F remains SHORT 5. 5.23 F DECIDES TO DELIVER 5 FUTURES C ACCEPTS DELIVERY OF 5 CONTRACTS. The actual delivery is now scheduled for June 23.

80. 79 CLEARINGHOUSE PROFIT/LOSS = ZERO* LONG PRICE SHORT PRICE TOTAL PROFIT A 10 $20 10 $22 $20,000 B 10 $21.5 10 $20 -$15,000 C 10 $21 10 $21.5 $5,000 10 $20 -$10,000 D 25 $22.5 25 $21 -$37,500 E 10 $22 10 $21 -$10,000 F 10 $21 25 $22.5 $15,000 10 $20 $25,000 TOTAL -$7,500 C TAKES DELIVERY 5 PAYS $21 : -$105,000 F DELIVERS 5 RECEIVES $22.5 : $112,500 $7,500 TOTAL 0 * This calculation accounts for buying and selling only. It does not account for cash movements resulting from the daily marking-to-market process. THE ACTUAL PROFITS AND LOSSES OF MARKET PARTICIPANTS ARE ACCUMULATED IN THE MARGIN ACCOUNTS.

81. 80 The following exhibits illustrate the activity in the margin account of each of the traders focusing only on cash flow resulting from the daily marking-to-market process. Thus, possible margin calls are ignored. PARTY A: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.16 LONG 10 $20 Initial margin LONG 10 $21 +$10,000 LONG 10 5.17 SHORT 10 $22 +$10,000 0 TOTAL $20,000 A’s profit is = $20,000 PARTY B: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.16 SHORT 10 $20 Initial margin SHORT 10 $21 -$10,000 SHORT 10 5.17 $22 -$10,000 SHORT 10 5.20 LONG 10 $21.5 +$5,000 0 TOTAL -$15,000 B’s loss is = $15,000

82. 81 PARTY C: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.16 LONG 25 $21 $21 Initial margin LONG 25 5.17 $22 +$25,000 5.20 SHORT 10 $21.5 -$5,000 $21.5 -$7,500 LONG 15 5.21 $20.5 -$15,000 LONG 15 5.22 SHORT 10 $20 -$5,000 $20 -$2,500 LONG 5 5.23 TAKE DELIVERY OF 5,000 BARRELS for $20/bbl -$100,000 0 C’s total loss up to and and including 5.22 is $10,000. Note that the 5 contracts that were delivered has accumulated the following amount over the period: 5.17 (5,000)($1) = $5,000 5.20 (5,000)(-$.5) = -$2,500 5.21 (5,000)(-$1) = -$5,000 5.22 (5,000)(-$.5) = -$2,500 5.23 (5,000)(-$20) = -$100,000 Payment upon delivery TOTAL………….-$105,000 The five contracts have accumulated total payment of $105,000. Observe: $105,000/5,000 = $21/bbl AS PER THE INITIAL COMMITMENT.

83. 82 PARTY D: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.16 SHORT 25 $21 Initial margin SHORT 25 $21 0 SHORT 25 5.17 $22 -$25,000 SHORT 25 5.20 LONG 25 $22.5 -$12,500 0 TOTAL -$37,500 D’s total loss is = $37,500 PARTY E: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.17 LONG 10 $22 Initial margin LONG 10 $22 0 LONG 10 5.20 $21.5 -$5,000 LONG 10 5.21 SHORT 10 $21 -$5,000 0 TOTAL -$10,000 E’s total loss is = $10,000

84. 83 PARTY F: DATE ACTION PRICE SETTLE CASH FLOW POSITION 5.20 SHORT 25 $22.5 Initial margin SHORT 25 $21.5 +$25,000 5.21 LONG 10 $21 +$5,000 $20.5 +$15,000 SHORT 15 5.22 LONG 10 $20 +$5,000 $20 +$2,500 SHORT 5 5.23 DELIVER 5,000 BARRELS for $20/bbl +$100,000 0 F’s total profit up to and including 5.22 is $52,500. Note that the 5 contracts that were delivered has accumulated the following amount over the period: 5.20 (5,000)($1) = $5,000 5.21 (5,000)($1) = $5,000 5.22 (5,000)($.5) = $2,500 5.23 (5,000)($20) = $100,000 Payment upon delivery TOTAL…………..$112,500 The five contracts that party F delivers accumulated a total of $112,500. Observe: $112,500/5,000 = $22.5/bbl AS PER INITIAL COMMITMENT.

85. 84 THE MARKET PARTICIPANTS: TRADERS OF FUTURES MAY BE CLASSIFIED BY THEIR GOALS: SPECULATORS: WILL OPEN A RISKY FUTURES POSITION FOR EXPECTED PROFITS. ARBITRAGERS: WILL OPEN SIMULTANEOUS FUTURES AND CASH POSITIONS IN ORDER TO MAKE AN ARBITRAGE PROFIT. HEDGERS:WILL OPEN A FUTURES POSITION IN ORDER MINIMIZE OR ELIMINATE ALL PRICE RISK.

86. 85 SPECULATORS: TAKE RISK FOR EXPECTED PROFIT. ON THE MARKET FLOOR, WE FIND EXCHANGE MEMBERS WHO TRADE FOR THEIR ON ACCOUNTS. THESE ARE SPECULATORS. SCALPERS: LARGE POSITIONS SMALL PRICE MOVEMENTS NEVER STAY OPEN OVERNIGHT DAY TRADERS:OPEN A POSITION IN THE MORNING CLOSE AT THE CLOSE OF THE SAME DAY. POSITION TRADERS: HOLD OPEN POSITIONS FOR LONGER PERIODS THEY USUALLY OPEN SPREAD POSITIONS. OUTRIGHT SPECULATION: GO LONG or GO SHORT A SPREAD: LONG CONTRACT 1 and simultaneously SHORT CONTRACT 2

87. 86 PROFIT IN SPREADS: MISALIGNMENT OF TWO DIFFERENT FUTURES PRICES CROSS COMMODITY SPREAD: SHORT JUNE CRUDE OIL CONTRACT LONG JUNE HEATING OIL CONTRACT CROSS EXCHANGE SPREAD LONG WHEAT CBT SHORT WHEAT KCB TIME OR, CALENDAR SPREAD: LONG CONTRACT MONTH 1, SAY JUNE SHORT CONTRACT MONTH 2, SAY SEPT. CALENDAR SPREAD SPREAD = F 0,t 1 - F 0,t 2 SPREAD = JUNE FUTURES - SEPT FUTURESPRICE

88. 87 How to open a calendar spread? Rule 1: If the spread between two contracts narrows, a profit will occur if the lower-priced contract has been purchased and the higher-priced contract sold. A loss occurs if the lower-priced contract is sold and the higher-priced contract is purchased. Rule 2: If the spread between two contracts widens, a profit will occur if the lower-priced contract has been sold and the higher priced contract purchased. A loss occurs if the lower- priced contract is purchased and the higher priced contract is sold.

89. 88 THEREFORE in deciding which contracts to buy and sell: Rule 1:If the spread is expected to narrow: SELL THE SPREAD! i.e.,buy the low priced contract and sell the high priced contract Rule 2:If spread is expected to widen: BUY THE SPREAD! i.e.,buy the high priced contract and sell the low priced contract.

90. 89 CALENDAR SPREAD THE SPECULATOR EXPECTS THE SPREAD TO NARROW ACTION : SELL THE SPREAD July December Heating Oil Heating OilSpread Initial Position buy $.80 sell $.92 + $.12 Terminal Position sell $.84, (.65) buy $.89, (.89) - $.05, (.24) gain $.04 gain $.03 net gain $.07 ( -.12 loss) sell $1.00 buy $1.05 -$.05 IN GALLONS: July DecemberSpread Initial Position buy 42,000 gal. sell 42,000 gal. $.12 $.80/gal $.92/gal value, $33,600 value, $38,640 Terminal Position sell 42,000 gal. buy 42,000 gal. $.05 $.84/gal. $.89/gal. value, $35,280 value, $37,380 gain =.04 x gain = $.03 x 42,000 = $1,680 42,000 = $1,260 net gain = $.07 x 42,000 = $2,940 TO TERMINATE THE POSITION: BUY THE SPREAD.

91. 90 PURE ARBITRAGE PROFIT: A PROFIT MADE 1. WITHOUT EQUITY and 2. WITHOUT ANY RISK.

92. 91 ARBITRAGE WITH FUTURES: SPOTFUTURES MARKET Buy the Sell futures product Or Sell the Buy futures product short

93. 92 ARBITRAGE:BUY AND SELL THE SAME COMMODITIY SIMULTANEOUSLY IN TWO DIFFERENT MARKETS FOR A (RISK-FREE) SURE PROFIT, WITHOUT ANY INVESTMENT. THE CLASSICAL EXAMPLE: S O, NY = $.9 /GALLON OF HEATING OIL S O, LONDON = $.8/GALLON OF HEATING OIL COST= $.05/GALLON. ARBITRAGE: BUY IN LONDON-80 CENTS/GALLON SELL IN NY+90 CENTS/GALLON SHIP TO NY- 5 CENTS/GALLON ARBITRAGE PROFIT: 5 CENTS/GALLON NO INVESTMENT IS REQUIRED! NO RISK IS TAKEN ! & MARKETS MUST ADJUST

94. 93 ARBITRAGEIN PERFECT MARKETS CASH -AND-CARRY NOW:1. BORROW CAPITAL 2. BUY IN THE SPOT MARKET AND CARRY IT TO DELIVERY 3. SELL FUTURES AGAINST THE STORED COMMODITY AT MATURITY:3. DELIVER THE STORED COMMODITY AGAINST THE SHORT FUTURES. 1. REPAY THE LOAN REVERSE CASH-AND-CARRY 1. SELL COMMODITY SHORT IN THE SPOT MARKET NOW:2. INVEST THE PROCEEDS IN GOVERNMENT SECURITIES 3. OPEN A LONG FUTURES POSITION AT MATURITY:2. COLLECT CAPITAL FROM INVESTMENT IN THE GOVERNEMENT SECURITIES 3. TAKE DELIVERY AGAINST LONG FUTURES POSITION. 1. CLOSE THE SHORT SPOT POSITION.

95. 94 EXAMPLE CASH - AND - CARRY ON AUG 15, 2001 SPOT CRUDE OIL$ 20/ bbl = S O AUGUST 2002 FUTURES$ 23/ bbl = F O, AUG 02 ANNUAL RATE10 % = CC 20e.1 = 22.10342 < 23 = F O, AUG 02 TRANSACTION t = 0CASH FLOW BORROW $20,000 FOR 1 YR AT 10% +20,000 BUY 1,000 BARRELS OF CRUDE -20,000 SELL ONE AUGUST 02 WTI FUTURES 0 0 t = 1 (AUGUST 2002) DELIVER THE 1,000 BARRELS TO CLOSE THE SHORT FUTURES POSITION +23,000 REPAY THE LOAN: -22,103.42 SURE PROFIT: 897.58 NOTICE:NO EQUITY IS USED and NO RISK IS TAKEN

96. 95 EXAMPLE REVERSE CASH - AND - CARRY ON AUG 15, 2001 SPOT CRUDE OIL$ 20 / bbl = S O AUGUST 2002 FUTURES$ 22/ bbl = F O, AUG 02 ANNUAL RATE10 % = CC 20e.1 = 22.10342 > 22 = F O, AUG 02 TRANSACTION t = 0CASH FLOW SELL 1,000 BARRELS SHORT +20,000 LEND $20,000 FOR 1YR AT 10% -20,000 BUY ON AUGUST 1997 FUTURES 0 0 t = 1 (AUGUST 2002) COLLECT 20,000e.1 +22,103.42 TAKE DELIVERY OF 1,000 BARRELS -22,000.00 DELIVER 1,000 TO CLOSE THE SHORT SPOT POSITION 0 +103.42 SURE PROFIT: 103.42 NOTICE:NO EQUITY IS USED and NO RISK IS TAKEN

97. 96 IN THE ABSENCE OF ARBITRAGE OPPORTUNITIES F 0, T = S 0 (1 + COST-OF-CARRY) IN OUR EXAMPLE: THE SPOT PRICE IS $20/bbl. THEREFORE, THE THEORETICAL FUTURES PRICE SATISFIES: F O, AUG 02 = 20e.1 = $22.10342 /bbl ANY OTHER FUTURES PRICE WILL LEAD TO ARBITRAGE OPPORTUNITIES

98. 97 ARBITRAGE IN THE REAL WORLD IMPEDIMENTS TRANSACTION COSTS DIFFERENT BORROWING AND LENDING RATES MARGINS REQUIREMENTS RESTRICTED SHORT SALES AN USE OF PROCEEDS STORAGE LIMITATIONS *BID - ASK SPREADS **MARKING - TO - MARKET *BID - THE HIGHEST PRICE ANY ONE IS WILLING TO BUY AT NOW ASK - THE LOWEST PRICE ANY ONE IS WILLING TO SELL AT NOW. **MARKING - TO - MARKET: YOU MAY BE FORCED TO CLOSE YOUR POSITION BEFORE ITS MATURITY.

99. 98 FOR THE CASH - AND - CARRY: BORROW AT THE BORROWING RATE: C B BUY SPOT FOR: S ASK SELL FUTURES AT THE BID PRICE: F(BID). PAY TRANSACTION COSTS ON: BORROWING BUYING SPOT SELLING FUTURES PAY CARRYING COST PAY MARGINS

100. 99 FOR THE REVERSE CASH - AND - CARRY SELL SHORT IN THE SPOT FOR: S BID. INVEST THE FACTION OF THE PROCEEDS ALLOWED BY LAW: f 0 ≦ f ≦ 1. LEND MONEY AT THE LENDING RATE: C L LONG FUTURES AT THE ASK PRICE: F(ASK). PAY TRANSACTION COST ON: SHORT SELLING SPOT LENDING BUYING FUTURES PAY MARGIN

101. 100 With these market realities, a new no-arbitrage condition emerges: B L < F < B U. F = B U F = B L time As long as F fluctuates between the upper and lower bounds there are no arbitrage profits.

102. 101 ARBITRAGEIN IMPERFECT MARKETS CASH -AND-CARRY NOW:1. BORROW CAPITAL 2. BUY IN THE SPOT MARKET AND CARRY IT TO DELIVERY 3. SELL FUTURES AGAINST THE STORED COMMODITY AT MATURITY:3. DELIVER THE STORED COMMODITY AGAINST THE SHORT FUTURES. 1. REPAY THE LOAN REVERSE CASH-AND-CARRY 1. SELL COMMODITY SHORT IN THE SPOT MARKET NOW:2. INVEST THE PROCEEDS IN GOVERNMENT SECURITIES 3. OPEN A LONG FUTURES POSITION AT MATURITY:2. COLLECT CAPITAL FROM INVESTMENT IN THE GOVERNEMENT SECURITIES 3. TAKE DELIVERY AGAINST LONG FUTURES POSITION. 1. CLOSE THE SHORT SPOT POSITION.

103. 102 Prove the following bounds on a futures price: S t, BID (1 - k)[1 + f(R L )] < F t,T < S t,ASK (1 + k)(1 + R B ) Where: S t, is the commodity’s spot price today, t. Note that you buy at the ASK price and sell at the BID price. F t,T is today’s futures price for delivery at T. For trading futures purposes, assume that F is used for buying and selling. That is, no BID or ASK price. k is the transaction cost associated with trading the spot commodity. k is a percentage of the price per unit. R L and R B are the annual lending and borrowing rates, respectively. f is the fraction of the proceeds from the commodity’s short sale that the arbitrageur may use. Note that the remainder, 1 - f must remain in the arbitrageur’s escrow account.

104. 103 S 0,BID (1 - T)[1 + f(C L )] < F 0, t < S 0,ASK (1 + T)(1 + C B ) EXAMPLE S 0, ASK = $20.50 / bbl S 0, BID = $20.25 / bbl C B = 12 % C L = 8 % T = 3 % $20.25(.97)[1+f(.08)]<F 0,t < $20.50(1.03)(1.12) $19.6425 + f($1.57) < F 0,t < $23.6488 DEPENDING ON f, ANY FUTURES PRICE BETWEEN THE TWO LIMITS WILL LEAVE NO ARBITRAGE OPPORTUNITIES. THE CASH-AND-CARRY WILL COST $23.6488/bbl. THE REVERSE CASH-AND-CARRY WILL COST 19.6425 + f(1.62). IF f=0.5 THE LOWER BOUND IS $20.45. IN THE REAL MARKET, f = 1, FOR SOME LARGE ARBITAGE FIRMS AND THEIR LOWER BOUND IS $21.26. THUS, IT IS CLEAR THAT THERE ARE DIFFERENT ARBITRAGE BOUNDS APPLICABLE TO DIFFERENT INVESTORS. THE TIGHTER THE BOUNDS, THE GREATER ARE THE ARBITRAGE OPPORTUNITIES.

105. 104 HEDGERS: HEDGERS TAKE FUTURES POSITIONS IN ORDER TO ELIMINATE PRICE RISK. THERE ARE TWO TYPES OF HEDEGES A LONG HEDGE TAKE A LONG FUTURES POSITION IN ORDER TO LOCK IN THE PRICE OF AN ANTICIPATED PURCHASE AT A FUTURE TIME A SHORT HEDGE TAKE A SHORT FUTURES POSITION IN ORDER TO LOCK IN THE SELLING PRICE OF AN ANTICIPATED SALE AT A FUTURE TIME. ANTICIPATORY HEDGES ARE HEDGES, LONG OR SHORT, THAT HEDGERS OPEN IN ANTICIPATION OF THE COMMODITY SPOT PRICE INCREASE IN THE FUTURE.

106. 105 FUTURES and CASH PRICES: AN ECONOMICS MODEL SPECULATORS:WILL OPEN RISKY FUTURES POSITIONS FOR EXPECTED PROFITS. HEDGERS: WILL OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE ALL PRICE RISK. ARBITRAGERS: WILL OPEN SIMULTANEOUS FUTURES AND CASH POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

107. 106 Demand for LONG futures positions by long HEDGERS Long hedgers want to hedge all of their risk exposure if the settlement price is less than or equal to the expected future spot price. c b a OdOd 0Quantity of long positions Long hedgers want to hedge a decreasing amount of their risk exposure as the premium of the settlement price over the expected future spot price increases. F t (k) Exp t [S t+k ]

108. 107 Supply of SHORT futures positions by short HEDGERS. Short hedgers want to hedge a decreasing amount of their risk exposure as the discount of the settlement price below the expected future spot price increases. f e d QSQS 0 Quantity of short positions Short hedgers want to hedge all of their risk exposure if the settlement price is greater than or equal to the expected future spot price. F t (k) Exp t [S t + k ]

109. 108 Equilibrium in a futures market with a preponderance of long hedgers. D S D QdQd 0 Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Supply schedule Demand schedule Premium QSQS

110. 109 Equilibrium in a futures market with a preponderance of short hedgers. S D QdQd 0 Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Supply schedule Demand schedule Discount D QSQS

111. 110 Demand for long positions in futures contracts by speculators. 0Quantity of long positions F t (k) Exp t [S t + k ] Speculators will not demand any long positions if the settlement price exceeds the expected future spot price. Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price. c b a

112. 111 Supply of short positions in futures contracts by speculators. 0 Quantity of short positions F t (k) Exp t [S t + k ] Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price Speculators will not supply any short positions if the settlement price is below the the expected future spot price f e d

113. 112 Equilibrium in a futures market with speculators and a preponderance of short hedgers. S D Q d Q E Q s 0Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Increased supply from speculators Discount D Increased demand from speculators

114. 113 Equilibrium in a futures market with speculators and a preponderance of long hedgers. S D 0Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Increased supply from speculators Premium D QEQE Increased demand from speculators

115. 114 Equilibrium in the spot market 0 Quantity of the asset F t (k); S t F t (k) e Premium QEQE Spot demand Excess supply of the asset when the spot market price is S t } Spot supply Exp t [S t + k ]

116. 115 Equilibrium in the futures market 0 Net quantity of long positions held by hedgers and speculators F t (k) Exp t [S t + k ] F t (k) e Premium Q } Excess demand for long positions by hedgers and speculators when the settlement price is F t (k) e Schedule of excess demand by hedgers and speculators

117. 116 HEDGING IS ONE COMPONENT OF CORPORATE FINANCIAL POLICY BY HEDGING THE FIRM MAY: * LOWER EXPECTED TRANSACTION COST * REDUCE THE PROBABILITY OF BANKRUPCY *SIGNAL TO CREDITORS THAT FIRM IS SAFER * REDUCE EXPECTED TAX LIMITATIONS * LOWER COST OF AGENCY CONTROL PROBLEMS * BENEFIT MANAGERS DIRECTLY

118. 117 Example: The Tax story Taxes and the Gain from Hedging: Consider an oil company whose assets consist solely of 1 million barrels of oil reserves that the firm intends to extract in one year at a cost of $25 per barrel. The current futures price for oil is $30 per barrel, and the oil price in one year has an equal chance of being $25 or $35 per barrel. For simplicity, assume that the current futures price equals the expected future spot price. The firm faces a 30 percent income tax rate and has a $1 million tax credit that it can apply up to the amount of income taxes paid.

119. 118 If the firm does not hedge, its after-tax profits under each oil price scenario will be: I. $25 per Barrel Before-tax profits = ($25 - $25)(1M) = $0.0 million Income tax= $0.0 million After-tax profits= $0.0 million The firm pays no taxes, because its taxable income is zero. It loses the $1 million tax credit. II. $35 per Barrel Before-tax profits= ($35 - $25)(1M) = $10.0 million Income tax = (.30)($10M)-$1M= $2.0M After-tax profits= $8.0 million The firm pays only $2M in taxes, because it fully utilizes its tax credit of $1M. The firm’s expected after-tax profits in one year are (0.5) ($0.0M)+ (0.5) ($8.0M) = $4.0M

120. 119 If the firm hedges with a short position in oil futures, its after-tax profits under the two oil price scenarios will be: Before-tax profits= ($30 - $25)(1M) = $5.0 million Income tax= (0.30)($5M) – 1M = $0.5 million After-tax profits = $4.5 million The expected after-tax profit is greater for the hedged firm than for the non hedged firm. The $0.5 million difference is exactly equal to the non hedged firm’s expected loss of the $1 million tax credit. The hedged firm always utilized its tax credit fully, so its value is higher than that of the non hedged firm.

121. 120 In general, the effect of hedging when tax credits and deductions are available is Unhedged Hedged Expected loss { firm } = { firm } - {of credit and } value value deductions The benefit of hedging when tax benefits could be lost will be mitigated if firms can carry tax credits and deductions forward and backward in time. Further, firms that will surely have ample income to use all of their credits and deductions will gain little value form hedging due to this tax effect.

122. A LONG HEDGE LONG FUTURES IN ORDER TO HEDGE THE PRODUCT PURCHASE TO BE MADE AT A LATER DATE. I.E.,, LOCK IN THE PURCHASE PRICE. RECALL: THERE ARE TWO TYPES OF HEDGING: A SHORT HEDGE SHORT FUTURES IN ORDER TO HEDGE THE SALE OF THE PRODUCT TO BE MADE AT A LATER DATE. I.E., LOCK IN THE SALE PRICE

123. 122 NOTATIONS: F k,t =THE FUTURES PRICE AT TIME k FOR DELIVERY AT TIME t. k < tk = current time t = delivery time S k = THE SPOT PRICE AT TIME k. THE TERMS SPOT AND CASH ARE USED INTERCHANGEABLY.

124. 123 BASIS: AT ANY POINT IN TIME, k: BASIS k = SPOT PRICE k - FUTURES PRICE k NOTATIONALLY: B k = S k - F k,t k < t B t = S t - F t, t = 0 k = t. t is the nearest month of delivery which is at or following k. The latter equation indicates that the basis converges to zero on the delivery date. F t, t is the price of the commodity on date t for delivery and payment on date t. Hence, F t, t is the spot price on date t F t, t = S t.

125. 124 The relationship between the cash and the futures price over time: 1.The basis is the difference between two random variables. Thus, it varies in an unpredictable way. Over time, it narrows, widens and may change its sign. 2.The basis converges to zero at the futures maturity. 3.The basis is less volatile than either price 4.Futures and spot prices of any underlying asset, co vary over time. Although not in tandem and not by the same amount, these prices move up and down together most of the time, during the life of the futures. RESULT 4. IS THE KEY TO THE SUCCESS OF HEDGING WITH FUTURES!

126. 125 October November December FUTURES PRICE BASIS = [CASH - FUTURES] CASH PRICE EXPIRATION = DELIVERY 82 81 80 79 78 Convergence of Cash and Futures-Heating Oil CENTS PER GALLON

127. 126 We now prove that hedging is the transfer of outright PRICE RISK to BASIS RISK. Generally, the basis fluctuates less than both, the cash and the futures prices. Hence, hedging with futures reduces risk. B0B0 O Pr S0S0 F 0,t kt time B t = 0 BkBk SkSk

128. 127 A LONG HEDGE TIMECASHFUTURES 0DO NOTHINGLONG F 0,t k BUY S k SHORT F k,t t delivery ACTUAL PAYMENT = S k + F 0,t - F k,t = F 0,t + [S k - F k,t ] = F 0,t + BASIS k

129. 128 A SHORT HEDGE TIMECASHFUTURES 0 DO NOTHINGSHORT F 0,t k SELL S k LONG F k,t t delivery ACTUAL SELLING PRICE = S k + F 0,t - F k,t = F 0,t + [S k - F k,t ] = F 0,t + BASIS k

130. 129 The last two slides prove that for both types of hedge A SHORT HEDGE or A LONG HEDGE, The final cash flow to the hedger is: F 0,t + BASIS k Notice that this cash flow consists of two parts: the first - F 0,t – is KNOWN when the hedge is opened. The second part - BASIS k – is a random element. Conclusion: At time 0, the firm faces the cash-price risk. Upon opening a hedging position, the firm locks in the futures price, but it still remains exposed to the basis risk, because the basis at time k is random.

131. 130 B0B0 O Pr S0S0 F 0,t kt time B t = 0 BkBk SkSk We thus, proved that hedging amounts to the reducing the firm’s risk exposure because the basis is less risky that the spot price risk.

132. 131 HEDGE RATIOS Open a hedge. Questions: Long or Short? Delivery month? Commodity to use? How many futures to use? The number of futures in the position is determined by the HEDGE RATIO

133. 132 HEDGE RATIOS NAÏVE HEDGE RATIO: ONE - FOR - ONE QUANTITIY OF CASH POSITION QUANTITY IN ONE FUTURE Examples: *Intend to sell 50,000 bbls of crude oil. Short 50 NYMEX futures. *Intend to borrow $10M for ten years. Short 100 CBT T-bond futures. *Intend to buy 17,000 pounds of gold. Long 170 NYMEX futures

134. 133 OPTIMAL HEDGE RATIOS THE MINIMUM VARIANCE HEDGE RATIO GOAL: TO MINIMIZE THE RISK ASSOCIATED WITH VALUE CHANGE OF THE CASH - FUTURES POSITION. RISK IS MEASURED BY VOLATILITY. THE VOLATILITY MEASURE ISTHE VARIANCE OF THE VALUE CHANGE OBJECTIVE: FIND THE NUMBER OF FUTURES THAT MINIMIZES THE VARIANCE OF THE CHANGE OF THE HEDGED POSITION VALUE.

135. 134 THE MATHEMATICS S = CASH VALUE F = FUTURES PRICE N = NUMBER OF FUTURES EMPLOYED IN THE HEDGE. The initial and terminal hedged position values: V P0 = S 0 + NF 0,t V P1 = S 1 + NF 1,t The position value change:  Vp= V P1  V P0 = (S 1 + NF 1,t ) - (S 0 + NF 0,t ). Define:  S = S 1 - S 0 and  F = F 1,t - F 0,t, then:  V P =  S + N(  F).

136. 135 AGAIN,  V P =  S + N(  F) PROBLEM: GIVEN THE CASH AND FUTURES VALUE CAHNGES, FIND A NUMBER N*, SO AS TO MINIMIZE THE VOLATILITY OF THE CHANGE IN THE HEDGER’S COMBINED CASH – FUTURES POSITION VALUE.

137. 136 THE MATHEMATICS. VAR (  V P ) = VAR (  S) + VAR (N  F) + 2COV (  S ; N  F) = VAR (  S) + N 2 VAR(  F) + 2NCOV(  S ;  F). TO MINIMIZE {VAR(  V P )} Take it’s derivative with respect to N and equate it to zero: 2N*VAR (  F) + 2COV (  S;  F) = 0 N* = - COV(  S;  F) / VAR(  F)

138. 137 THE NUMBER OF FUTURES THAT MINIMIZES THE RISK OF THE HEDGED POSITION IS:

139. 138 To evaluate the risk of the position at its minimum level, substitute N* into the formula of the position’s value change variance: How to calculate N* in practice?

140. 139 DATA (SAY DAILY) n+1 DAYS.

141. 140 EXAMPLE: A company needs to buy 800,000 gallons of diesel oil in 2 months. It opens a long hedge using heating oil futures. An analysis of price changes ΔS and ΔF over a 2 month interval yield: SD(ΔS) = 0.025; SD(ΔF) = 0.033;ρ = 0.693. The risk minimizing hedge ratio: h = (.693)(0.025)/0.033 = 0.525. One heating oil contract is for 42,000 gallons, so purchase N* = (0.525)(800,000)/42,000 = 10 futures.

142. 141 EXAMPLE, continued: Notice that in this case, a NAÏVE HEDGE ratio would have resulted in taking a long position in: 800,000/42,000 = 19 futures. Taking into account the correlation between the spot price changes and the futures price changes, allows the use of only 10 futures. Of course, if the correlation and the standard deviations take on other values the risk-minimizing hedge ratio may require more futures than the naïve ratio.

143. 142 EXAMPLE: A company knows that it will buy 1 million gallons of jet fuel in 3 months. The company chooses to long hedge with heating oil futures. The standard deviation of the change in the price per gallon of jet fuel over a 3-month period is calculated as 0.04. The standard deviation of the change in the futures price over a 3-month period is 0.02 and the coefficient of correlation between the 3- month change in the price of jet fuel and the 3-month change in the futures price is 0.42. The optimal hedge ratio: H = (0.42)(0.04)/(0.02) = 0.84, And the risk-minimizing number of futures N* = (0.84)(1,000,000)/42,000 = 20.

144. 143 HEDGE RATIOS As we move from one type of underlying asset to another, we will use these hedge ratios as well as new ones to be developed later.

145. 144 Delivery month? Normally, the hedge is opened with futures for the delivery month closest to the firm operation date in the cash market or the nearest month beyond that date. The key factor here is the correlation between the cash and futures prices or price changes. Statistically, it is known that in most cases, the highest correlation is with the futures prices of the delivery month nearest to the cash activity.