1. Derivatives Hedging with Futures
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Identifying the exposure
Exposure: position to be hedged
Cash flow(s)
Future income Ex: oil/gold producer
Future expense Ex: user of commodity
Value
Asset Ex: asset manager
Liability Ex: financial intermediary
General formulation:
Exposure = M  S
with: M = quantity, size (M > 0 asset, income M < 0 liability, expense)
S = market price
Derivatives 03 Hedging with Futures
22/09/2018

3. Setting up the hedge Futures position:
Number of contracts n (n>0 long hedge – n<0 short hedge)
 Size of one contract N
 Futures price F
Hedge = n  N  F
Perfect hedge: choose n so that value of hedged position does not change if S changes
Derivatives 03 Hedging with Futures
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4. Hedge ratio To achieve ∆V = 0 If M >0 : n <0 short hedge
If M<0 : n>0 long hedge
Derivatives 03 Hedging with Futures
22/09/2018

5. Perfect hedge Assume F and S are perfectly correlated:
then: h = - β and
Derivatives 03 Hedging with Futures
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6. Basis risk Basis = Spot price of asset – Futures prices (S-F)
Spot price S1 S2
Futures price F1 F2
Basis b1= S1 –F1 b2 = S2 – F2
Cash flow at time t2:
Long hedge: -S2 + (F2 – F1) = – F1 – b2
Short hedge: +S2 + (F1 – F2) = + F1 + b2 t1 t2
known at time t1
Derivatives 03 Hedging with Futures
22/09/2018

7. Minimum variance hedge
Real life more complex:
1. asset to be hedged might differ from underlying the futures contract
2. basis (S –F) might vary randomly
More general specification:
Choose n to minimize the variance of ∆V
Derivatives 03 Hedging with Futures
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8. Some math Take derivative and set it equal to 0: Solve for n:
Derivatives 03 Hedging with Futures
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9. Hedging Using Index Futures
Stock index futures: futures on hypothetical portfolio tracked by index.
Size = Index × Value of 1 index point
Example: S&P 500 (CME) - $250 × index
To hedge the risk in a portfolio the number of contracts that should be shorted is
where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract
Derivatives 03 Hedging with Futures
22/09/2018

10. Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.)
Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)
Derivatives 03 Hedging with Futures
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11. Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million
Beta of portfolio is 1.5
What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?
Derivatives 03 Hedging with Futures
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12. Changing Beta
What position is necessary to reduce the beta of the portfolio to 0.75?
What position is necessary to increase the beta of the portfolio to 2.0?
Derivatives 03 Hedging with Futures
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13. Rolling The Hedge Forward
We can use a series of futures contracts to increase the life of a hedge
Each time we switch from 1 futures contract to another we incur a type of basis risk
Derivatives 03 Hedging with Futures
22/09/2018