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6Metallgesellschaft

6Metallgesellschaft - Hedging with futures Review A case...

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Slide 6-1 Hedging with futures Review A case study: Metallgesellschaft
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Slide 6-2 Review: Futures hedging Tailing Maturity mismatch Asset mismatch Basis risk
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Slide 6-3 Hedging with futures: tailing Gains and losses on futures contracts are settled daily. Cash flow at T from a short futures position: where F t =F t,T denotes the futures price at time t , and F T =S T (no maturity mismatch). How can we turn this into a perfect hedge? Solution : adjust h 0 , h 1 , ..., h T-1 such that h 0 F 1 (1+r) T = h 1 F 1 (1+r) T-1 ! h 0 =h 1 /(1+r) h 1 F 2 (1+r) T-1 = h 2 F 2 (1+r) T-2 ! h 1 =h 2 /(1+r) ... h T-1 F T-1 (1+r) = h T F T-1 ! h T-1 =h T /(1+r) and h T = exposure to hedge h 0 ( F 0 " F 1 )(1 # r ) T # h 1 F 1 " F 2 $ % (1 # r ) T " 1 # ... # h T " 1 F T " 1 " F T $ %
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Slide 6-4 Hedging with futures: maturity mismatch 2 cases: Maturity of futures position is longer than the date of the exposure Maturity of futures position is shorter than the date of the exposure If the futures expire too late (1st case): lack of convergence between the futures price and the spot price at the date of the exposure. If the futures expire too soon (2nd case): we must roll-over futures positions; the first futures position is used to hedge the payoff of subsequent futures positions.
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Slide 6-5 The maturity of the futures is too long Suppose that we want to hedge an exposure of 100.000 oz of gold with a 12 month futures position that will be closed out after 6 months. S 0 = $736/oz, & g =0.3% p.a., r=4% p.a., daily compounding. Initial size of the position? 1. T 124 trading days ( T F 251 trading days ! h 0 S T F T , T F 1 (1 # r ) T 2. F T , T F S T (1 # r ) T F " T (1 # & g ) T F " T S T (1 # 0.04 /251) 251 " 124 (1 # 0.003/251) 251 " 124 S T 1.019 3. h 0 1 1.019 1 (1 # 0.04 /251) 124 0.962
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Slide 6-6 The maturity of the futures is too short Suppose that we want to hedge an exposure of 100.000 oz of gold with by rolling over a 6 month futures contract. S 0 = $736/oz, & g =0.3% p.a., r=4% p.a., daily compounding. Initial size of the position? 1. T 251 trading days ) T F 124 trading days ! h 0 F T F , T S T F 1 (1 # r ) T 2. F T F , T S T F (1 # r ) T " T F (1 # & g ) T " T F S T F (1 # 0.04 /251) 251 " 124 (1 # 0.003/251) 251 " 124 S T F 1.019 3. h 0 1.019 1 (1 # 0.04 /251) 251 0.979
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Slide 6-7 Basis risk What if future interest rates are unknown? Expected profit from marking to market ! 0. Future forward premia are unknown. ! Risk from maturity mismatch cannot be eliminated through tailing. Basis risk: Asset mismatch: Maturity mismatch: Asset mismatch and maturity mismatch: S T # ( F 0 " F * ) F 0 # ( S T " F * ) Basis ! # $ % $ S T # ( ˆ F 0 " ˆ F T ) ˆ F 0 # ( S T " ˆ F T ) Basis ! # $ % $ S T # ( ˆ F 0 " ˆ F T ) ˆ F 0 # ( S T " ˆ F * ) Basis ! # $ % $
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Slide 6-8 Hedging in the presence of basis risk Goal: to minimize the variance of the final payoff. Optimal hedge ratio: The optimal hedge ratio is the beta of the exposure with respect to the underlying asset of the futures contract.
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