Hedging Strategies Using
Futures
Chapter 3
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.1
Hedging issues
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Hedging vs speculating – hedging to reducing particular
risk
Perfect hedge – complete elimination of risk
Issues in hedging strategies
 Short or long hedge?
 Size of the hedge?
 Which futures contract to be used?
Hedge-and-forget strategy vs dynamic hedging
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.2
Long & Short Hedges
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A long futures hedge is appropriate when
you know you will purchase an asset in
the future and want to lock in the price
A short futures hedge is appropriate
when you know you will sell an asset in
the future & want to lock in the price
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.3
Arguments in Favor of Hedging
Companies should focus on the main
business they are in and take steps to
minimize risks arising from interest
rates, exchange rates, and other market
variables
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.4
Arguments against Hedging

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Shareholders are usually well diversified
and can make their own hedging decisions
It may increase risk to hedge when
competitors do not
Explaining a situation where there is a loss
on the hedge and a gain on the underlying
can be difficult
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.5
Convergence of Futures to Spot
(Hedge initiated at time t1 and closed out at time t2)
Futures
Price
Spot
Price
Time
t1
t2
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.6
Basis Risk


Basis is the difference between spot
price & futures price
Basis risk arises because of the
uncertainty about the basis when the
hedge is closed out
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.7
Long Hedge

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Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future purchase of an asset
by entering into a long futures contract
Cost of Asset=S2 – (F2 – F1) = F1 + Basis
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.8
Short Hedge


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Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future sale of an asset by
entering into a short futures contract
Price Realized=S2+ (F1 – F2) = F1 + Basis
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.9
A Short Hedge

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May 15: An oil producer will have 1 million
barrels of crude oil to sell on August 15.
Quotes:
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Spot price of crude oil: $60 per barrel
August futures price: $59 per barrel
May 15: The producer sells 1,000 August
contracts on crude oil
August 15: Closes out futures position
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.10
Short hedge – cont.
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Suppose the spot price on August 15 is
$55 per barrel.
Producers gain on futures contract =
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
$59 - $55 = $4
Total amount realized = 55 + 4 = $59
Note that the futures price equals spot
price on Aug 15, due to convergence
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.11
Long Hedge


Jan 15: A copper fabricator will require
100,000 pounds of copper on May 15.
Quotes Jan 15:
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Spot price = 340 cents per pound
May futures = 320 cents per pound
Jan 15:The producer takes a long position
on copper futures
May 15: Closes out the futures position
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.12
Long hedge – cont.


Suppose the copper price on May 15 is
325 cents per pound.
Gain on futures = $3.25 – 3.20 = $.05/lbs
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Total = .05 x 100,000 = $5000
Total price paid = $3.25 - .05 = $3.20/lbs
OR: S2 – F2 + F1
OR: S2 - (F2 – F1) =F1+ (S2 - F2)
OR: F1+ basis
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.13
Basis risk in a short hedge
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March 1: US exported expects to receive
¥50 million at the end of July. September
futures on ¥ is currently @ ¢0.7800/ ¥.
Strategy: Short four September contracts
on Yen; close out position end of July.
Suppose end of July:

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Spot price = ¢0.7200/ ¥
September futures = ¢0.7250/ ¥.
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.14
Basis risk in a short hedge – cont.

There are two ways of arriving at the net
exchange rate after hedging:
S2 - (F2 – F1) = F1+ (S2 - F2)
(1) Spot price in July - gain on futures

0.7200 - (.7250 - .7800) = .7750
(2) Futures price in March + Basis in July
.7800 + (.7200 - .7250) = .7800 - .0050
= .7750
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.15
Basis risk in a long hedge
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Jun 8: A company will need 20,00 barrels of crude oil in
October or November. The current December oil futures
price is $68 per barrel.
Strategy: long 20 December contracts @ $68/bbl. Close
out position when ready to buy.
Suppose the company is ready to purchase oil on
November 10. At the time:
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Spot price = $75/bbl
December futures @ $72
Basis = 75 – 72 = $3
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.16
Basis risk in a long hedge

There are two ways of arriving at the net
cost of crude after hedging:
S2 - (F2 – F1) = F1+ (S2 - F2)
(1) Spot price in Nov. - gain on futures

75 - (72 - 68) = 75 - 4 = $71
(2) Futures price in March + Basis in July
68 + (75 - 72) = 68 + 3 = $71
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.17
Choice of Contract


Choose a delivery month that is as close as possible to,
but later than, the end of the life of the hedge
When there is no futures contract on the asset being
hedged, choose the contract whose futures price is most
highly correlated with the asset price. There are then 2
components to basis:
 Choice of the asset underlying futures
 Choice of the delivery month
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.18
Optimal Hedge Ratio
Proportion of the exposure that should optimally be
hedged is
s
hr S
sF
where
 sS is the standard deviation of DS, the change in
the spot price during the hedging period,
 sF is the standard deviation of DF, the change in
the futures price during the hedging period
 r is the coefficient of correlation between DS and
DF.
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.19
Optimal Number of Contracts
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QA = Size of position being hedged (units)
QF: Size of one futures contract (units)
N* : Optimal number of futures contracts
*
* QA
N h
QF
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.20
Trailing the hedge

In futures contract adjustment must be
made to the optimal number of contracts
for the daily settlement:
*
* VA
N h
VF

Where VA and VF is the dollar value of the
position and of one futures contract.
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.21
Stock Index Futures
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CBT: DJIA based on 30 blue chip companies;
price weighted index; valued @ $10 x index
CBT: Mini-DJIA , valued @ $5 x index
CME: S&P500 index, market capitalization
weighted index; valued@ $250 x index
CME: Mini S&P500, valued @ $50 x index
CME: Nasdaq 100; valued @ $100 x index
CME: Mini-Nasdaq 100; valued @ $20 x index
NYBOT: Russell 1000; valued @ $500 x index
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.22
Hedging Using Index Futures
(Page 62)
To hedge the risk in a portfolio the
number of contracts that should be
shorted is
P
b
F
where P is the value of the portfolio,
b is its beta, and F is the current
value of one futures (=futures price
times contract size)
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.23
Reasons for Hedging an Equity
Portfolio
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
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 outpeform the
market.)
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.24
Example
S&P 500 index = 1,000
Futures price of S&P 500 is 1,010
Size of portfolio is $5,050,000
Risk-free interest rate =4% per annum
Dividend yield on index = 1% per annum
Beta of portfolio is 1.5
What position in futures contracts on the
S&P 500 is necessary to hedge the
portfolio?
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.25
No of contracts needed

One contract is on $250 times the index
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F = 250 x 1010 = 252,500
Number of futures contracts to short:
P
5,050,000
b  1.5 
 30
F
252,500
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.26
Gain/Loss
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
Suppose the index turns out to be 900 in
three months, and the futures price is 902.
Gain from short futures position:
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30 x (1010 -902) x 250 = $810,000
Loss on the index = 10%

Considering dividend yield loss on net loss on
the index would be 10% - 1%/4 = 9.75%
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.27
Value of the portfolio

According to CAPM, the expected return on the portfolio
is:
 E[rp] = rf + β(rm –rf)
 When market index return is -9.75%
 E[rp] = 1 + 1.5(-9.75 – 1) = -15.125%
 Or the expected value of the portfolio is:
 5,050,000 x (1 – 15.125%) = $4,286,187
 Total value of the portfolio include gain on the short
hedge:
 4,286,187 + 810,000 = $5,096,187 close to the
original value
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.28
Changing Beta

What position is necessary to reduce the beta of
the portfolio to 0.75?
P
5,050,000
( b  b )  (1.5  0.75) 
 15
F
252,500
What position is necessary to increase the beta
of the portfolio to 2.0?
5,050,000
* P
( b  b )  (1.5  2.0) 
 10
F
252,500
*


i.e., 10 long contracts
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.29
Rolling The Hedge Forward
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
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
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
3.30