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Hedging Strategies Using Futures

Hedging Strategies Using Futures. Chapter 4. HEDGERS OPEN POSITIONS IN THE FUTURES MARKET IN ORDER TO ELIMINATE THE RISK ASSOCIATED WITH THE PRICE OF THE UNDERLYING ASSET IN THE SPOT MARKET. There are two ways to determine whether to open a short or a long hedge:. 1. A LONG HEDGE.

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Hedging Strategies Using Futures

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  1. Hedging Strategies Using Futures Chapter 4

  2. HEDGERS OPEN POSITIONS IN THE FUTURES MARKET IN ORDER TO ELIMINATE THE RISK ASSOCIATED WITH THE PRICE OF THE UNDERLYING ASSET IN THE SPOT MARKET.There are two ways to determine whether to open a short or a long hedge:

  3. 1. A LONG HEDGE A SHORT HEDGE OPEN A LONG FUTURES POSITION IN ORDER TO HEDGE THE PRODUCT PURCHASE TO BE MADE AT A LATER DATE. In this case, it is known that a purchase will be made at a future date. THW HEDGWR LOCKS IN THE PURCHASE PRICE. OPEN A SHORT FUTURES POSITION IN ORDER TO HEDGE THE SALE OF THE PRODUCT TO BE MADE AT A LATER DATE. In this case, it is known that a sale will be made at a future date. THE HEDGER LOCKS IN THE SALE PRICE

  4. 2. A LONG HEDGE A SHORT HEDGE OPEN A LONG FUTURES POSITION WHEN THE FIRM HAS A SHORT SPOT POSITION. OPEN A SHORT FUTURES POSITION WHEN THE FIRM HAS A LONG SPOT POSITION.

  5. Arguments in Favor of Hedging • Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

  6. Arguments against Hedging • Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult. • Shareholders are usually well diversified and can make their own hedging decisions.

  7. NOTATIONS: t< T t = current time; T = delivery time F t,T = THE FUTURES PRICE AT TIME t FOR DELIVERY AT TIME T. St = THE SPOT PRICE AT TIME t. Some times we use t = 0 for the initial futures purchase or sale. .

  8. DELIVERY k = is the date on which the hedger conducts the firm spot business and simultaneously closes the futures position. This date is almost always before the delivery month. WHY? • From the first trading day of the delivery month, the SHORT can decide to send a delivery note. Any LONG with an open position may be served with this delivery note. • Often k is not in a delivery month.

  9. DEFINITION The basis is the difference between the asset’s spot and the futures price. BASISk = SPOT PRICEk - FUTURES PRICEk Notationally: Bk = Sk - Fk,T k < T. In general, one must specify a specific futures, i.e., a specific delivery month. Usually, however, it is understood that the futures is for the nearest month to delivery.

  10. A LONG HEDGE TIMESPOTFUTURES t Contract to buy LONG F t,T Do nothing k BUY Sk SHORT F k,T T delivery ACTUAL PAYMENT = Sk + Ft,T - Fk,T = Ft,T + [Sk - Fk,T] = Ft,T +BASISk

  11. A SHORT HEDGE TIMECASHFUTURES t Contract to sell SHORT Ft,T Do nothing k SELL Sk LONG Fk,T T delivery ACTUAL SELLING PRICE = Sk + Ft,T - Fk,T = Ft,T + [Sk - Fk,T] = Ft,T +BASISk

  12. Observe that in both cases, Long hedge and short hedge The cash flow for the hedger when the hedge is closed on date k, is: Ft,T +BASISk This cash flow consists of a known price Ft,T and a random value BASISk We return to this point later.

  13. THE KEY FOR THE SUCCESS OF HEDGING WITH FUTURES IS: the relationship between the cash and the futures price over time: Futures and spot prices of any underlying asset, co vary over time. Although not in perfect tandem and not by the same amount, these prices move up and down together most of the time, during the life of the futures.

  14. Example: A LONG HEDGE DateSpot marketFutures market t St = $400/unit Ft,T = $425/unit Contract to buy long one gold Gold on k. futures for delivery at T K Buy the gold Short one gold Sk = $416/unit futures for delivery at T. Fk,T = $442/unit T Amount paid: 416 + 425 – 442 = $399/unit or 425 + (416 – 442) = $399/unit

  15. Example: A SHORT HEDGE DateSpot marketFutures market t St = $400/unit Ft,T = $425/unit Contract to sell short one gold Gold on k, futures for delivery at T K Sell the gold Long one gold Sk = $384/unit futures for delivery at T. Fk,T = $412/unit T Amount received: 384 + 425 – 412 = $397/unit or 425 + (384 – 412) = $397/unit

  16. BASISk = SPOT PRICEk - FUTURES PRICEk Notationally: Bk = Sk - Fk,T k < T. But on the delivery date k = T: BT = ST - FT, 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. FT,T is the price of the commodity on date T for delivery and payment on date T. Hence, FT,T = ST .

  17. Convergence of Futures to Spot Futures Price Spot Price Futures Price Spot Price Time Time (a) (b)

  18. The relationship between the cash and the futures price over time: • 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. • The basis converges to zero at the futures maturity. • The basis is less volatile than either price.

  19. Basis Risk • Basis is the difference between spot & futures prices. • Basis risk arises because of the uncertainty about the basis when the hedge is closed out at time k. • We do know, however, that BT = 0 at delivery.

  20. Generally, the basis fluctuates less than both, the cash and the futures prices. Hence, hedging with futures reduces risk. Nonetheless, Basis risk exists in any hedge. Sk Pr Bk Ft,T St BT = 0 Bt time k T t

  21. We showed above that for both types of hedge A SHORT HEDGE or A LONG HEDGE, The final payment received or paid to the hedger is: Ft,T +BASISk This cash flow consists of two parts: the first, Ft,T is KNOWN when the hedge is opened. The second part - BASISk – is a random element. Conclusion: At time t, 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.

  22. We thus, proved that: hedging amounts to the reducing the firm’s risk exposure because the basis is less risky than the spot price risk. Sk Pr Bk Ft,T St BT = 0 Bt time k T t

  23. Choice of contract for hedging • Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge • When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. Then, there are two components to the basis: The basis associated with the asset underlying the futures and the spread between the two spot prices. Let S1 be the spot asset price. This is the asset that the hedger is trying to hedge. Let S2 be the spot price of the asset underlying the futures. This is a different asset.

  24. A CROSS HEDGE TIMECASHFUTURES t Contract to trade Ft,T Do nothing k Trade for S1K Fk,T T delivery PAY OR RECEIVE = S1K + Ft,T - Fk,T = Ft,T + [S2k - Fk,T] +[S1k - S 2k] = Ft,T +BASISk + SPREADK

  25. 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.

  26. 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

  27. HEDGE RATIOS; DEFINITION: NS = The number of units of the commodity to be traded in the SPOT market. NF = The number of units of the commodity in ONE FUTURES CONTRACT. n = The number of futures contracts to be used in the hedge. h = The hedge ratio.

  28. NAÏVE HEDGE RATIO: ONE - FOR - ONE

  29. Examples:NAÏVE HEDGE RATIO: ONE - FOR – ONE 1. Intends to sell NS = 50,000 barrels of crude oil. NF = 1,000 barrels Short n = 50,000/1,000 = 50 NYMEX futures. 2. Intend to borrow $10M for ten years. Hedge with CBT T-bond futures. NS = 10,000,000; NF = 100,000. Short n = 10,000,000/100,000 = 100 CBT T-bond futures. 3. Intend to buy NS = 200,000 bushels of wheat. CBT wheat futures: NF = 5,000. Long n = 200,000/5,000 = 40 CBT futures.

  30. Other hedge ratios. Suppose that the relationship between the spot and futures prices over time is: SpotFutures  $1  $2  $1  $0.5 Clearly, the Naïve hedge ratio is not appropriate in these cases.

  31. OPTIMAL HEDGE RATIOS THE MINIMUM VARIANCE HEDGE RATIO OBJECTIVE: TO MINIMIZE THE RISK ASSOCIATED WITH THE HEDGE. RISK IS MEASURED BY: VOLATILITY. THE VOLATILITY MEASURE IS THE VARIANCE OBJECTIVE: FIND THE NUMBER OF FUTURES, n, THAT MINIMIZES THE VARIANCE OF THE CHANGE OF THE HEDGED POSITION VALUE.

  32. THE MATHEMATICS St = Spot market price. t = 0 The hedge opening date. t = 1 The hedge closing date. T = The delivery date. F0,T = The futures price on date 0 for delivery at T. n = The number of futures used in the hedge. h = The hedge ratio. NF = The number of units of the asset in one contract. NS = The number of units of the asset to be traded spot on t = 1.

  33. The initial and terminal hedged position values: VP0 = S0NS +nNFF0,T VP1 = S1NS +nNFF1,T The position value change: (Vp) = VP1 - VP0 = (S1NS +nNFF1,T) - (S0NS +nNFF0,T) = NS(S1- S0) +nNF(F1,T - F0,T).

  34. AGAIN: (VP) = NS(S1- S0) +nNF(F1,T - F0,T). (VP) = NS[(S1- S0) +nNF/NS(F1,T - 0,T)] (VP) = NS[(S1- S0) +h(F1,T - F0,T)] PROBLEM: (VP) is a random variable because the prices at t=1 are unknown. Find h* so as to minimize the risk associated with (VP).

  35. VAR(VP) = = NS2 VAR [(S1- S0) +h(F1,T - F0,T)] = NS2[VAR(S)+VAR(hF)+2COV(S;hF)] = NS2 [VAR(S) + h2VAR(F) +2hCOV(S;F)]. TO MINIMIZE VAR(VP) take it’s derivative with respect to h and equate it to zero: 2h*VAR (F) + 2COV(S; F) = 0. h* = - COV(S;F)/VAR(F)

  36. THE MINIMUM RISK HEDGE RATIO IS:

  37. The negative sign only indicates that the SPOT and the FUTURES positions are in opposite directions. If the hedger is short spot, the hedge is long. If the hedger is long short, the hedge is short.

  38. DATA (SAY DAILY) n+1 DAYS.

  39. EXAMPLE 1: 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: (ΔS) = 0.025; (ΔF) = 0.033; ρ(ΔS;ΔF) = 0.693. The risk minimizing hedge ratio: h* = (.693)(0.025)/0.033 = 0.525. One NYMEX heating oil contract is for 42,000 gallons, so Long n* = (0.525)(800,000)/42,000 = 10 futures.

  40. EXAMPLE 1, continued: Notice that in this case, a NAÏVE HEDGE ratio would have resulted in taking a long position in: n* =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 The minimum variance hedge ratio and thus, the optimal number of futures: 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.

  41. EXAMPLE 2: 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.

  42. EXAMPLE 3. A Hedging example for copper: Date: OCT03 FEB04 AUG04 FEB05 AUG05 Spot: 72.00 69.00 65.00 77.00 88.00 Futures For Delivery: MAR 2004 72.30 69.10 SEP 2004 72.80 70.20 64.80 MAR 2005 71.60 70.70 64.30 76.70 SEP 2005 69.50 68.90 64.20 76.50 88.20 Today is OCT 2003. A US firm has a contract to purchase 1,000,000 pounds of copper in FEB 04, AUG04, FEB05 and AUG05. The firm decides to hedge these purchases with NYMEX copper futures. One NYMEX copper futures is for 25,000 pounds of copper. The firm decides to use h* = .7.

  43. We now analyze three potential Hedging strategies. HEDGING POLICY I: The hedge ratio is h* = .7. No other restrictions. The firm uses a STRIP . That is, a sequence of futures that are equally spaced, each one hedging one spot future trade.

  44. DateSPOT MARKETFUTURES MARKETFFUTURES POSITIONS Oct 03 NOTHING Long 28 MAR 2004 72.30 long 28 MAR 2004 Long 28 SEP 200472.80long 28 SEP 2004 Long 28 MAR 2005 71.60 long 28 MAR 2005 Long 28 SEP 2005 69.50 long 28 SEP 2005 Feb 04 buy 1M units 69.00 short 28 MAR 04 69.10 long 28 SEP 2004 long 28 MAR 2005 long 28 SEP 2005 Aug 04 buy 1M units 65.00 short 28 SEP 04 64.80 long 28 MAR 2005 long 28 SEP 2005 Feb 05 buy 1M units 77.00 short 28 MAR 05 76.70 long 28 SEP 2005 Aug 05 buy 1M units 88.00 short 28 SEP 05 88.20 NO POSITION The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75 The average price for the hedged strategy: (.3)69 + (.7)(69 + 72.30 – 69.10) = 71.24 (.3)65 + (.7)(65 + 72.8 – 64.8) = 70.60 (.3)77 + (.7)(77 + 71.6 – 76.7) = 73.43 (.3)88 + (.7)(88 + 69.5 – 88.2) = 74.98 72.5625

  45. Hedging strategy II: The hedge ratio is h* = .7 The firm will not use futures with delivery months which are more than 13 months hence. Thus, the firm uses a STACK. All the futures needed to be stacked will be stacked to the latest delivery months futures used.

  46. Hedging strategy II: DateSPOT MARKETFUTURES MARKETFFUTURES POSITIONS Oct 03 NOTHING Long 28 MAR 2004 72.30 long 28 MAR 2004 Long 84 SEP 2004 72.80 long 84 SEP 2004 Feb 04 buy 1M units 69.00 short 28 MAR 04 69.10 short 56 SEP 04 70.20 long 28 MAR 2005 long 56 MAR 05 70.70 long 28 SEP 2005 Aug 04 buy 1M units 65.00 short 28 SEP 04 64.80 long 28 SEP 2004 short 28 MAR 05 64.30 long 56 MAR 2005 long 28 SEP 05 64.20 Feb 05 buy 1M units 77.00 short 28 MAR 05 76.70 long 28 SEP 2005 Aug 05 buy 1M units 88.00 short 28 SEP 05 88.20 NO POSITION The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75 The average price for the hedged strategy: (.3)69 + (.7)(69 + 72.30 – 69.10) +(1.35)(72.80 – 70.20) = 74.75 (.3)65 + (.7)(65 + 72.8 – 64.80) + (.7)( 70.70 – 64.30) = 75.08 (.3)77 + (.7)(77 + 70.7 – 76.7) = 72.80 (.3)88 + (.7)(88 + 64.2 – 88.2) = 71.20 73.4575

  47. Hedging strategy III. The hedge ratio is h* = .7 The firm uses the following stacking policy: On OCT 2003 Stack all the futures needed to be stacked onto the SEP 2004 position. On FEB 2004 Do not change your SEP2004 position. That is, keep the entire stack on the SEP2004 position. On AUG 2004 Enter into whatever futures positions the firm needs, which are not restricted and continue with them to the hedge termination date.

  48. Hedging policy III. DateSPOT MARKETFUTURES MARKETFFUTURES POSITIONS Oct 03 NOTHING Long 28 MAR 2004 72.30 long 28 MAR 2004 Long 84 SEP 2004 72.80 long 84 SEP 2004 Feb 04 buy 1M units short 28 MAR 04 69.10 long 84 SEP 2004 69.00 Aug 04 buy 1M units short 84 SEP 04 64.80 65.00 long 28 MAR 05 64.30 long 28 MAR 2005 long 28 SEP 05 64.20 long 28 SEP 2005 Feb 05 buy 1M units short 28 MAR 05 6.70 long 28 SEP 2005 77.00 Aug 05 buy 1M units short 28 SEP 05 88.20 NO POSITION 88.00 The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75 The average price for the hedged strategy: (.3)69 + (.7)(69 + 72.30 – 69.10) = 71.24 65 + 2.1)( 72.8– 64.80) = 81.80 (.3)77 + (.7)(77 + 64.3 – 76.7) = 68.32 (.3)88 + (.7)(88 + 64.2 – 88.2) = 71.20 73.14

  49. 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. • The two underlying assets that we analyze next are: • 1. Stock index futures. • Foreign currency futures. • In each case, we first describe the SPOT MARKET and • then analyze the FUTURES MARKET.

  50. STOCK INDEX FUTURES The first stock index futures began trading in 1982 on the KCBT. The underlying was the VALUE LINE INDEX. Soon after, the CBT, after losing its battle with the Dow Jones Co., started trading futures on the MAJOR MARKET INDEX, the MMI. Today, Stock Index Futures are traded one dozens of indexes.

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