Derivatives Hedging with Futures

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DerivativesHedging with Futures

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

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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 gt 0 asset, income M
  lt 0 liability, expense)
• S market price

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Setting up the hedge

• Futures position
• Number of contracts n (ngt0 long hedge nlt0 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

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Hedge ratio

• To achieve ?V 0
• Hedge ratio
• To achieve ?V 0

If M gt0 n lt0 short hedge If Mlt0 ngt0 long hedge
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Perfect hedge

• Assume F and S are perfectly correlated
• then h - ß and

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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
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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

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Some math
Take derivative and set it equal to 0
Solve for n
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Hedging Using Index Futures

• Stock index futures futures on hypothetical
  portfolio tracked by index.
• Size Index Value of 1 index point
• Example SP 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

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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.)

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Example

• Value of SP 500 is 1,000
• Value of Portfolio is 5 million
• Beta of portfolio is 1.5
• What position in futures contracts on the SP
  500 is necessary to hedge the portfolio?

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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?

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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