Unformatted text preview: ive full credit for tax losses, the tax code does not have an effect on the expected
aftertax profits of firms that have the same expected pretax profits, but different cashflow
variability.
Question 4.18. Auric Enterprises is using gold as an input. Therefore, it would like to hedge against price
increases in gold.
a)
The cost of this collar today is the premium of the purchased 440strike call ($2.49) less
the premium for the sold 400strike put. We calculate a cost of $2.49 − $2.21 = −$0.28 , which
means that Auric in fact generates a revenue from entering into this collar. 32 Chapter 4 Introduction to Risk Management b)
A good starting point are the values of part a). You see that both put and call are worth
approximately the same, therefore start shrinking the 440 – 400 span symmetrically until you get
a difference of 30, and then do some trial and error. This should bring you the following values:
The call strike is 435.52, and the put strike is 405.52. Both call and put have a premium of
$3.425
Question 4.20. a)
Since we know that the value of a call is decreasing in the strike, and we need to sell two
call options, the BlackScholes prices of which equal the 440strike call price, we know that we
have to look for a higher strike price. Trial and error results in a strike price of 448.93. The
premium of the 440strike call is $2.4944, and indeed the BlackScholes premium of the 448.93
strike call is $1.2472.
b) Profit diagram: 33 Chapter 5
Financial Forwards and Futures
Question 5.2.
a)
The owner of the stock is entitled to receive dividends. As we will get the stock only in
one year, the value of the prepaid forward contract is today’s stock price, less the present value
of the four dividend payments:
4 F0PT = $50 − ∑ $1e
, 3
− 0.06× i
12 = $50 − $0.985 − $0.970 − $0.956 − $0.942 = $50 − $3.853 = $46.147 i =1 b)
The forward price is equivalent to the future value of the prepaid forward. With an
interest rate of 6 percent and an expiration of the forward in one year, we thus have: F0,T = F0PT × e 0 .06×1 = $46.147 × e 0 .06×1 = $46.147 × 1.0618 = $49.00
,
Question 5.4. This question asks us to familiarize ourselves with the forward valuation equation.
a)
We plug the continuously compounded interest rate and the time to expiration in years
into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We
have: F0,T = S 0 × e r×T = $35 × e 0.05×0.5 = $35 × 1.0253 = $35.886
b) The annualized forward premium is calculated as:
annualized forward premium = 1 F0,T
ln
T S0 1 $35.50 = 0.5 ln $35 = 0.0284 c)
For the case of continuous dividends, the forward premium is simply the difference
between the riskfree rate and the dividend yield: S × e (r −δ )T = 1 ln 0
T
S0 1
1
= ln (e (r −δ )T ) = (r − δ )T
T
T
= r −δ annualized forward premium = 1 F0,T
ln
T S0 34 Chapter 5 Financial Forwards and Futures Therefore, we can solve:
0.0284
⇔ δ = 0.05 − δ
= 0.0216 The annualized dividend yield is 2.16 percent.
Question 5.6. a)
We plug the continuously compounded interest rate, the dividend yield and the time to
expiration in years into the valuation formula and notice that the time to expiration is 9 months,
or 0.75 years. We have: F0,T = S 0 × e (r −δ )×T = $1,100 × e (0.05−0.015 )×0.75 = $1,100 × 1.0266 = $1,129.26
b) We engage in a reverse cash and carry strategy. In particular, we do the following:
Description
Today
Long forward, resulting 0
from customer purchase
Sell short tailed position + S 0 e −δT
of the index
− S 0 e −δT
Lend S 0 e −δT
TOTAL
0 In 9 months
S T − F0,T − ST
S 0 × e (r −δ )T S 0 × e (r −δ )T − F0,T Specifically, we have:
Description
Today
Long forward, resulting 0
from customer purchase
Sell short tailed position $1,100 × .9888
of the index
= 1087.69
− $1,087.69
Lend $1,087.69 TOTAL 0 In 9 months
ST − $1,129.26 − ST
$1,087.69 × e 0.05×0.75
= $1,129.26
0 Therefore, the market maker is perfectly hedged. He does not have any risk in the future, because
he has successfully created a synthetic short position in the forward contract. 35 Part 2 Forwards, Futures, and Swaps c)
Description
Today
Short forward, resulting 0
from customer purchase
Buy tailed position in − S 0 e −δT
index
Borrow S 0 e −δT
S 0 e − δT
TOTAL
0 In 9 months F0,T − ST
ST
− S 0 × e (r −δ )T F0,T − S 0 × e (r −δ )T Specifically, we have:
Description
Today
Short forward, resulting 0
from customer purchase
Buy tailed position in − $1,100 × .9888
index
= −$1,087.69
$1,087.69
Borrow $1,087.69
TOTAL 0 In 9 months
$1,129.26 − S T
ST
− $1,087.69 × e 0.05×0.75
= −$1,129.26
0 Again, the market maker is perfectly hedged. He does not have any index price risk in the future,
because he has successfully created a synthetic long position in the forward contract that
perfectly offsets his obligation from the sold forward contract.
Question 5.8. First, we need to find the fair value of the forward price. We plug the continuously compounded
interest rate, the dividend yield and th...
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 Spring '14
 NguyenThiMaiLan
 Derivatives, simple strategies

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