Derivatives Hedging with Futures Professor Andr Farber Solvay
Derivatives Hedging with Futures Professor André Farber Solvay Business School Université Libre de Bruxelles Derivatives 03 Hedging with Futures
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 1/6/2022 Derivatives 03 Hedging with Futures 2
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 1/6/2022 Derivatives 03 Hedging with Futures 3
Hedge ratio If M >0 : n <0 short hedge • To achieve ∆V = 0 If M<0 : n>0 long hedge • Hedge ratio: • To achieve ∆V = 0 1/6/2022 Derivatives 03 Hedging with Futures 4
Perfect hedge • Assume F and S are perfectly correlated: • then: h = - β and 1/6/2022 Derivatives 03 Hedging with Futures 5
Basis risk • Basis = Spot price of asset – Futures prices t 1 (S-F) t 2 • Spot price • Futures price • Basis S 1 F 1 b 1= S 1 –F 1 • Cash flow at time t 2: • Long hedge: • Short hedge: -S 2 + (F 2 – F 1) = – F 1 – b 2 +S 2 + (F 1 – F 2) = + F 1 + b 2 S 2 F 2 b 2 = S 2 – F 2 known at time t 1 1/6/2022 Derivatives 03 Hedging with Futures 6
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 1/6/2022 Derivatives 03 Hedging with Futures 7
Some math Take derivative and set it equal to 0: Solve for n: 1/6/2022 Derivatives 03 Hedging with Futures 8
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 1/6/2022 Derivatives 03 Hedging with Futures 9
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. ) 1/6/2022 Derivatives 03 Hedging with Futures 10
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? 1/6/2022 Derivatives 03 Hedging with Futures 11
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? 1/6/2022 Derivatives 03 Hedging with Futures 12
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 1/6/2022 Derivatives 03 Hedging with Futures 13
- Slides: 13