Hedging Strategies Using
Futures
Chapter 3
Options, Futures, and Other
Derivatives, 7th Edition, Copyright ©
John C. Hull 2008
1

Hedge : A trade designed to reduce risk.
◦ Many of the participants in futures markets are
hedgers. Their aim is to use futures markets to
reduce a particular risk that they face.

A perfect hedge is one that completely
eliminates the risk.
◦ Perfect hedges are rare.
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Short Hedges
A short futures hedge is appropriate
when
1. The hedger already owns an asset and
expects to sell it at some time in the
future
 Example: A farmer who owns some
hogs and knows that they will be ready
for sale in two months.

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Short Hedges
2.

An asset is not owned right now but will be
owned at some time in the future
Example: A US exporter who knows that
he or she will receive euros in 3 months.
The exporter will realize a gain if the euro
increases in value r.t. the US dollar and
will sustain a loss if it decreases in value.
A short futures position offsets the risk.
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Long Hedges
A long hedge is appropriate when
1. a company knows it will have to
purchase a certain asset in the future
and wants to lock in a price now
 Example: In January 15, a copper
fabricator knows it will require 100,00
pounds of copper on May 15 to meet a
certain contract

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Long Hedges
2.

An investor has a short position and
wants to manage its risk
Example: An investor who has
shorted a certain stock. Part of the
risk faced by him is related to the
performance of the whole stock
market, and could be neutralize with
a long position in index futures
contracts.
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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
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Arguments against Hedging
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

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Convergence of Futures to Spot
(Hedge initiated at time t1 and closed out at time t2)
Futures
Price
Spot
Price
Spot
Price
Futures
Price
Time
t1
t2
Time
t1
t2
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Basis Risk
Basis is the difference between the
spot and futures price
 Basis risk arises because of the
uncertainty about the basis when
the hedge is closed out

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Long Hedge
We define
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
 If you hedge the future purchase of an
asset by entering into a long futures
contract then
Cost of Asset=S2 – (F2 – F1) = F1 + Basis

Basis = S2 – F2
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Short Hedge
Again we define
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
 If you hedge the future sale of an asset by
entering into a short futures contract then
Price Realized=S2+ (F1 – F2) = F1 + Basis

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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
◦ Close, because basis risk increases as the time
difference between the hedge expiration and the
delivery month increases
◦ Later, since if not, your asset will be exposed to
the risk of not being hedged
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.
This is known as cross hedging.
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Minimum Variance Hedge Ratio
Formula
Let h* be the proportion of the exposure that
should optimally be hedged
sS : the standard deviation of DS, the change in
the spot price during the hedging period
sF : the standard deviation of DF, the change in
the futures price during the hedging period
r: the coefficient of correlation between DS and
DF.
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Minimum Variance Hedge Ratio
Formula
N A : units of the original asset
N F : units of the asset underlying
h
NF
the futures
: the hedge ratio
NA
Y : The total amount realized on the asset
in Spot and Futures Market
Y  S 2 N A  ( F 2  F1 ) N F
 S 1 N A  ( S 2  S 1 ) N A  ( F 2  F1 ) N F
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MVHR continued
Set
D S  S 2  S1 ,
D F  F 2  F1
Then
Y  S1 N A  N A (D S  hD F )
Var Y  Var ( D S  h D F )
 s S  h s F  2 h rs S s F
2
d Var Y
dh
2
 2hs
2
F
2
 2 rs S s F  0  h  r
*
sS
sF
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Figure 3.3 Regression of change in spot price against change in
futures price
h* is the slope of this line
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Optimal Number of Contracts
QA: Size of position being hedged (units of the asset)
QF: Size of one futures contract (units of the asset )
N*: Optimal number of futures contracts for hedging
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Tailing the Hedge
A procedure for adjusting the number of futures
contracts used for hedging to reflect daily
settlement
VA : the dollar value of the position being hedged
VF : the dollar value of one futures contract (the
futures price times QF)
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Tailing the Hedge (continued)

Two ways of determining the number of
contracts to use for hedging are
◦ Compare the exposure to be hedged with the
value of the assets underlying one futures
contract
◦ Compare the exposure to be hedged with the
value of one futures contract (=futures price time
size of futures contract)

The second approach incorporates an
adjustment for the daily settlement of
futures
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Hedging An Equity Portfolio
To hedge the risk in a portfolio the
number of contracts that should be
shorted (N* ) is
P
b
F
where P is the value of the portfolio, b is
its beta, and F is the value of one futures
contract
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Example
Suppose that a futures contract with 4 months to
maturity is used to hedge the value of a portfolio
over the next 3 months in the following situation:
a.
b.
Value of S&P 500 index = 1,000
S&P 500 futures price = 1,010
Value of portfolio = $5,050,000
Risk-free interest rate = 4 % per annum
Dividend yield on index = 1 % per annum
Beta of portfolio = 1.5
What position should be taken to eliminate the exposure to
the market over the next 3 months?
Calculate the effect of the strategy if the level of the market
in 3 months is 900 and the futures price is 902.
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Answer
a. N*= β(P/F)= 1.5($505,0000 / $252,500)~ 30
(the number of contracts in which the manager should take short positions)
b. Y=N* (F2-F1) = 30 (1010 - 902) * $ 250 =$ 810,000
(the gain from the short futures position)
The loss on the index in the form of capital gain =
100(900 - 1000)/1000 = -10%
Return on the index in the form of dividend =1*(3/12) =0.25%
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Answer continued
µm = 0.25 - 10 = - 9.75%
µp = rf + β (µm- rf) = 1+1.5 (-9.75 – 1)= -15.125%
Expected Portfolio Value (EPV) = P (1+ µp)
= 505,0000(1 – 0.15125) = $ 4,286,187
Total value of position = EPV + Y
= 428,6187 + 810,000 = $ 5096187
Total gain = 5096187 – 5050000= $ 46187
Options, Futures, and Other
Derivatives, 7th Edition, Copyright ©
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Changing Beta
What position is necessary to reduce
the beta of the portfolio to 0.75?
2. What position is necessary to
increase the beta of the portfolio to
2.0?
1.
1. N* = (β – β*) (P/F) = (1.5 – 0.75) * (5000000/(250 * 1000)) = 15 (short position)
2. N* = (β* – β) (P/F) = (2 – 1.5) * (5000000/(250 * 1000)) = 10 (long position)
Options, Futures, and Other
Derivatives, 7th Edition, Copyright ©
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Hedging Price of an Individual
Stock
Similar to hedging a portfolio
 Does not work as well because only the
systematic risk is hedged
 The unsystematic risk that is unique to the
stock is not hedged

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Why Hedge Equity Returns
May want to be out of the market for a
while. Hedging avoids the costs of selling
and repurchasing the portfolio
 Suppose stocks in your portfolio have an
average beta of 1.0, but you feel they have
been chosen well and will outperform the
market in both good and bad times.
Hedging ensures that the return you earn is
the risk-free return plus the excess return of
your portfolio over the market.

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Rolling The Hedge Forward
(page 64-65)
We can use a series of futures contracts to
increase the life of a hedge
 Each time we switch from one futures
contract to another we incur a type of
basis risk

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