# 6 Hedging Using Futures

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1 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 1 6 Hedging Using Futures 6.1 Types of hedges using futures Two types of hedge: short and long.

2 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 2 1. Short hedge Take a short hedge position in the futures market. Appropriate when someone expects to sell an asset he already owns and wants to guarantee the price. Example: I m selling 1 million barrels of crude oil. The spot price for a barrel is \$19 per barrel and the 3-month futures price is \$18.75 per barrel. Spot price in three months proves to be \$17.50: I gain \$18.75-\$17.50=\$1.25 per barrel from the futures but I m selling the oil for \$1.25 less per barrel. I end up getting \$18.75 per barrel. Spot price in three months proves to be \$19.50: I lose \$19.50-\$18.75=\$0.75 per barrel from the futures but I m selling the oil for \$0.75 more per barrel. I end up getting \$18.75 per barrel.

3 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 3 2. Long hedge Take a long position in the futures market. Appropriate for someone who expects to buy an asset and wants to guarantee the price.

4 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures Basis and basis risk Perfect hedge does not always exist The asset we are trying to hedge may not be exactly the same as the asset underlying the futures. The time at which we sell the asset (which could be random) might not be exactly be the same as the delivery date of the futures.

5 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 5 Basis 1. Basis spot price of asset to be hedged - futures price of asset being used for the hedge b t = S t F t If the asset being hedged and used for the hedge are the same, then the basis will be zero at the expiration of the futures contract.

6 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 6 2. Intuition of why it is useful: Consider a short hedger who will sell an asset at time T and who takes a short futures position at t 0 < T for delivery at some time t 1 T. Income at T is: (i) price of asset at time T : S T (ii) profit on futures position: (F t0,t 1 F T,t1 )e r(t 1 T) F t0,t 1 F T,t1 if t 1 T is small, as is usually the case. Total income at T S T + (F t0,t 1 F T,t1 ) = F t0,t 1 + (S T F T,t1 ) = F t0,t 1 + b T The basis b T collects all the uncertain terms and so collects all the risk into one number.

7 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 7 3. Same formula for basis applies to a long hedge Consider someone who knows he will buy an asset at T and takes a long position at t 0 Total payment at T is (i) spot price S T (ii) loss on hedge F t0,t 1 F T,t1 (approximately, as before) total payment = S T + (F t0,t 1 F T,t1 ) = F t0 + b T as before. Note that b T 0 if the futures contract expires at T because then S T = F T.

8 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 8 Basis risk If b T were known at t 0, we would have a perfect hedge (because if b T is known, then b T is fixed. This means that S T and F T must always change by equal amounts, leaving income unchanged). When b T is unknown at time t 0, the uncertainty about the period T income, captured by the uncertainty about the value of b T, hence called basis risk.

9 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 9 Another source of risk: different assets held and contracted The asset in the futures contract may not be the same as the asset Define S price of asset in futures contract Total income/payment becomes S T + F t0,t 1 F T,t1 = F t0,t 1 + (ST F T,t1 ) + (S T ST) where ST F T,t1 = basis if the asset being hedged were the same as the asset in the futures contract. S T ST = difference in price between the two assets.

10 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures Minimum variance hedge ratio Hedge ratio The size of the position in futures contracts relative to the exposure (i.e., the value of the asset being hedged). More simply, the number of units of asset in futures contracts per unit of asset being hedged.

11 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 11 Riskiness of a short hedger s portfolio 1. Value of portfolio in arbitrary time period t N A number of units of asset being hedged. h hedge ratio (ratio of the size of the position taken in futures contracts to the size of the exposure). Then value of the portfolio is the sum of: (i) N A S t = spot value of asset holding (ii) N A h(k F t,t )e r(t t) = value of the futures contracts owned where T is expiration date of futures contract total portfolio value = N A [S t + h(k F t,t )e r(t t) ].

12 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures Value at time when delivery of asset is made Denote this time by t 1, with t 1 T Suppose the futures contract was written in period t 0 < t 1, so that K = F t0. Value of portfolio in period t 1 is N A [S t1 + h(f t0,t F t1,t)e r(t t 1) ]

13 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 13 Approximate value of portfolio if T t 1 is small, as is generally the case, then e r(t t 1) 1 and we have N A [S t1 + h(f t0,t F t1,t)] which is the expression we will use. Per extended unit value * Each unit of the asset is accompanied by h units of a futures contract * Call S t1 + h(f t0,t F t1,t) the value of an extended unit of the asset. * All the risk arises from this value, not from N A, so we analyze the behavior of S t1 + h(f t0,t F t1,t).

14 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures Riskiness of the value At t 0, when the futures contract is written, the values of S t1 and F t1,t are unknown random variables. The change in the value of the extended unit is [S t1 + h(f t0,t F t1,t)] [S t0 + h(f t0,t F t0,t)] = [S t1 + h(f t0,t F t1,t)] [S t0 + 0] = (S t1 S t0 ) + h(f t0,t F t1,t) = S h F where S S t1 S t0 and F F t1,t F t0,t.

15 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 15 This change is a random variable with variance equal to V ar [ S h F] = V ar[ S] + h 2 V ar[ F] 2hCov[ S, F] = σ 2 S + h 2 σ 2 F 2hσ SF = σ 2 S + h 2 σ 2 F 2h ρ σ S σ F Note that σ S, σ F, ρ are functions of t 1. Notice that the same expression applies to a long hedge * Change in value of payment to be made is * Same variance expression (S t1 S t0 ) + h (F t0,t F t1,t) } {{ } loss on futures contract

16 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 16 Choose h to minimize the variance V ar h = 2hσ 2 F 2ρσ S σ F = 0 for a minimum Which gives h = ρ σ S σ F = regression coefficient from regressing S on F

17 ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 17 Optimum number of contracts Define: * N A = size of position being hedged (i.e., number of units of the asset being hedged). * Q F = size of one futures contract (i.e., number of units of asset in one futures contract). * N = optimal number of futures contracts. Value of N is N = h N A Q F

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