Financial Derivatives Tutorial 1: Futures and

Forwards - Solutions

Daniele Massacci

King’s College London

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Question 1
I Question 1: A cattle farmer expects to sell in 3 months
120,000 lbs of live cattle. CME live-cattle futures are for
delivery of 40,000 lbs of cattle. How can the farmer hedge
with the contract? What are the advantages and the
disadvantages of hedging?

I Answer:
I Hedging strategy: Short 3 contracts of 3 months maturity. If
cattle price falls, futures gain o¤sets cattle sale loss. If cattle
prices rises, cattle sale gain will be o¤set by futures loss.
I Pros: Futures cost nothing (no premium) and reduce outcome
uncertainty to almost zero.
I Cons: Farmer has no potential gain from favorable cattle price
movements.

I Reference: If interested, check CME group Live Cattle

Futures Quotes here.
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Question 2

I Question 2: Consider a long position in a 1 year forward

contract on a stock that pays no dividend. The current price
of the stock is S0 = $40, the risk-free rate is r = 10% on a
yearly basis with continuous compounding.
1. What is the forward price? What is the initial value of the
forward contract?
2. Six months later, the price of the stock is S6 = $45, and the
risk free rate remains r = 10% on a yearly basis with
continuous compounding. What is the forward price? What is
the value of the forward contract?

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Question 2
I Answer:
1. In general,
F0 = S 0 e rT
and
rT
f = S0 Ke ,
where K is the delivery price and Ke rT is the present value of
the delivery price. In our case, S0 = $40, r = 10%/yr and
T = 1. It follows that

F0 = S 0 e rT = $40 e 0.1 1
' $44.21

and
rT rT
f = S0 Ke = S0 F0 e = S0 S0 e rT e rT
=0

since at the beginning of the contract K = F0 .

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Question 2

I Answer - continued:
2. After six months we have S6 = $45, r = 10%/yr and
T = 0.5. It follows that

F6 = S6 e rT = $45 e 0.1 0.5

' $47.31

and
0.1 0.5
f = $45 $44.21e ' $2.95.

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Question 3

I Question 3: The expected return on the S&P 500 is 12%

and the risk-free rate is 5%. What does the CAPM tell us if
the expected return on the investment has beta (β) equal to
(a) 0.2, (b ) 0.5 and (c ) 1.4?

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Question 3
I Answer: According to the CAPM

E (return) = r + β (rm r) ,

where r is the risk-free rate and rm is the return on the

market, with r = 0.05 and rm = 0.12. It follows that:
(a) If β = 0.2 then

E (return) = 0.05 + 0.2 (0.12 0.05) = 0.064;

(b ) If β = 0.5 then

E (return) = 0.05 + 0.5 (0.12 0.05) = 0.085;

(c ) If β = 1.4 then

E (return) = 0.05 + 1.4 (0.12 0.05) = 0.148.

Notice that E (return) monotonically increase in β.

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Question 4

I Question 4: A company wants to hedge exposure to a new

fuel whose price changes have a 0.6 correlation (ρAF ) with
gasoline futures price changes. The company loses $1 million
for each 1 cent increase in price per gallon of a new fuel over
the next three months. The new fuel’s price change has a
standard deviation 50% greater than that of gasoline futures
prices. If gas futures are used to hedge, what is the hedge
ratio? What is the company’s exposure in gallons of new fuel?
What position in gallons should the company take in gas
futures? How many gas futures contracts should be traded if
each contract is for 42,000 gallons?

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Question 4
I Answer: The hedge ratio is de…ned as
ρσe
h = ,
σf
where σe and σf are the standard deviations of exposure and
futures, respectively. From the question, we know that
ρ = 0.6 and σe = 1.5σf . We thus have:
1. The hedge ratio is
ρσe 0.6 1.5σf
h = = = 0.9.
σf σf

2. The company’s exposure is

Loss $1, 000, 000

Exposure = = = 100, 000, 000 gallons
Price Change $0.01/gallons

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Question 4

I Answer-continued:
3. The company should take a long position on

0.9 100, 000, 000 = 90, 000, 000 gallons gasoline futures.

4. The optimal number of futures contracts is

h QA
N = .
QF

From the question, we know that QF = 42, 000. Therefore,

90, 000, 000
N = = 2142.9 ' 2143.
42, 000

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Question 5

I Question 5: The standard deviation of monthly changes in

live cattle spot prices (in cents per pound) is σA = 1.2. The
standard deviation of monthly changes in live cattle futures
prices is σF = 1.4. The correlation between futures prices
changes and spot prices changes is ρAF = 0.7. On October 15
a beef producer committed to purchasing 200, 000 lbs of live
cattle on November 15. The producer wants to use the
December futures to hedge. Each contract is 40, 000 lbs
cattle. What strategy should the producer follow?

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Question 5
I Answer: Recall that the optimal hedge ratio h is equal to
ρ σA 0.7 1.2
h = = = 0.6.
σF 1.4
The beef producer takes a long position on

200, 000 0.6 = 120, 000

lbs worth of futures contracts. These are equal to

120, 000
N= =3
40, 000
futures contracts. Therefore, the beef producer takes a
long position on N = 3 December futures contracts closing
out the position on November 15.

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Question 6
I Question 6: The risk-free interest rate is r = 7% per annum
with continuous compounding, and the dividend yield on a
stock index is q = 3.2% per annum. The current index value
is S0 = $150. What is the 6-month futures price?

I Answer: The futures price is

F0 = S 0 e (r q) T
,

with S0 = $150, r = 0.07, q = 0.032 and T = 0.5: therefore

F0 = $150 e (0.07 0.032 ) 0.5

= $150 e 0.038 0.5

= $152.88.

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Question 7

I Question 7: Suppose the risk-free rate is r = 10% per

annum with continuous compounding, and the dividend yield
on a stock index is q = 4% per annum. The current index
value is S0 = $400 and the futures price for a contract
deliverable in 4 months is $405. What arbitrage opportunities
does this create?

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Question 7

I Answer: The futures price is

F0 = S 0 e (r q) T
,

with S0 = $400, r = 0.10, q = 0.04 and T = 1/3: therefore

F0 = $400 e (0.10 0.04 ) 1/3

= $400 e 0.02
= $408.08.

Therefore, the current market price of $405 is less than

F0 = $408.08: an investor could buy the futures, short the
shares in the underlying asset, earn the interest on cash from
short-sales of stocks at 10% per annum for the next 4 months.

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Question 8

I Question 8: The correlation coe¢ cient between an

investment portfolio and the market is de…ned as
σAM
ρAM = ,
σA σM
where σAM is the covariance between the portfolio and the
market, and σA and σM are the covariance of the portfolio
and of the market, respectively. Show that the minimum
variance hedge ratio h when trying to cross-hedge an asset is
equivalent to β.

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Question 8

I Answer: We have
σA
h =ρ
σM
and since
σAM
ρ=
σA σM
then
σA σ σ σ
h =ρ = AM A = AM = β.
σM σA σM σM σ2M

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Question 9

I Question 9: "Options and futures are zero-sum games."

What do you think it meant by this statement?

I Answer:
I The statement means that the gain (loss) to the party with the
short position is equal to the loss (gain) to the party with the
long position. In total, the gain to all parties is zero.

I Typically investments in the underlying are NOT zero-sum

games, equities can experience growth, bonds pay out more
than the original loan principals.

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