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Unformatted text preview: e 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 −δ )×T = $1,100 × e (0.05 −0.02 )×0.5 = $1,100 × 1.01511 = $1,116.62
a)
If we observe a forward price of 1,120, we know that the forward is too expensive,
relative to the fair value we have determined. Therefore, we will sell the forward at 1,120, and
create a synthetic forward for 1,116.82, making a sure profit of $3.38. As we sell the real
forward, we engage in cash and carry arbitrage: 36 Chapter 5 Financial Forwards and Futures Description
Today
Short forward
0
Buy tailed position in − $1,100 × .99
index
= −$1,089.055
$1,089.055
Borrow $1,089.055
TOTAL
0 In 9 months
$1,120.00 − ST
ST
− $1,116.62
$3.38 This position requires no initial investment, has no index price risk, and has a strictly positive
payoff. We have exploited the mispricing with a pure arbitrage strategy.
b)
If we observe a forward price of 1,110, we know that the forward is too cheap, relative to
the fair value we have determined. Therefore, we will buy the forward at 1,110, and create a
synthetic short forward for 1116.62, thus making a sure profit of $6.62. As we buy the real
forward, we engage in a reverse cash and carry arbitrage:
Description
Today
Long forward
0
Sell short tailed position in $1,100 × .99
index
= $1,089.055
− $1,089.055
0 Lend $1,089.055
TOTAL In 9 months
ST − $1,110.00
− ST
$1,116.62
$6.62 This position requires no initial investment, has no index price risk, and has a strictly positive
payoff. We have exploited the mispricing with a pure arbitrage strategy.
Question 5.10. a)
We plug the continuously compounded interest rate, the forward price, the initial index
level and the time to expiration in years into the valuation formula and solve for the dividend
yield:
F0,T = S 0 × e (r −δ )×T
⇔ F0,T
S0 = e (r −δ )×T ⇔ F
ln 0,T
S
0 = (r − δ ) × T ⇔ δ =r− 1 F0,T
ln
T S0 ⇒ δ = 0.05 − 1 1129.257 ln = 0.05 − 0.035 = 0.015
0.75 1100 37 Part 2 Forwards, Futures, and Swaps Remark: Note that this result is consistent with exercise 5.6., in which we had the same forward
prices, time to expiration etc.
b)
With a dividend yield of only 0.005, the fair forward price would be: F0,T = S 0 × e (r −δ )×T = 1,100 × e (0.05−0.005 )×0.75 = 1,100 × 1.0343 = 1,137.759
Therefore, if we think the dividend yield is 0.005, we consider the observed forward price of
1,129.257 to be too cheap. We will therefore buy the forward and create a synthetic short
forward, capturing a certain amount of $8.502. We engage in a reverse cash and carry arbitrage:
Description
Today
Long forward
0
Sell short tailed position in $1,100 × .99626
index
= $1,095.88
Lend $1,095.88
TOTAL c) − $1,095.88
0 In 9 months
ST − $1,129.257
− ST
$1,137.759
$8.502 With a dividend yield of 0.03, the fair forward price would be: F0,T = S 0 × e (r −δ )×T = 1,100 × e (0.05−0.03 )×0.75 = 1,100 × 1.01511 = 1,116.62
Therefore, if we think the dividend yield is 0.03, we consider the observed forward price of
1,129.257 to be too expensive. We will therefore sell the forward and create a synthetic long
forward, capturing a certain amount of $12.637. We engage in a cash and carry arbitrage:
Description
Today
In 9 months
Short forward
0
$1,129.257 − ST
Buy tailed position in − $1,100 × .97775 ST
index
= −$1,075.526
$1,075.526
$1,116.62
Borrow $1,075.526
TOTAL
0
$12.637
Question 5.12. a)
The notional value of 10 contracts is 10 × $250 × 950 = $2,375,000 , because each index
point is worth $250, we buy 10 contracts and the S&P 500 index level is 950.
With an initial margin of 10% of the notional value, this results in an initial dollar margin of
$2,375,000 × 0.10 = $237,500 . 38 Chapter 5 Financial Forwards and Futures b)
We first obtain an approximation. Because we have a 10% initial margin, a 2% decline in
the futures price will result in a 20% decline in margin. As we will receive a margin call after a
20% decline in the initial margin, the smallest futures price that avoids the maintenance margin
call is 950 × .98 = 931 . However, this calculation ignores the interest that we are able to earn in
our margin account.
Let us now calculate the details. We have the right to earn interest on our initial margin position.
As the continuously compounded interest rate is currently 6%, after one week, our initial margin
has grown to:
$237,500e 0.06× 1
52 = $237,774.20 We will get a margin call if the initial margin falls by 20%. We calculate 80% of the initial
margin as:
$237,500 × 0.8 = $190,000 10 long S&P 500 futures contracts obligate us to pay $2,500 times the forward price at expiration
of the futures contract.
Therefore, we have to solve the following equation:
$237,774.20 + (F1W − 950) × $2,500 ≥ $190,000 ⇔ $47774.20 ≥ −(F1W − 950) × $2,500 ⇔ 19.10968 − 950 ≥ − F1W ⇔ F1W ≥ 930.89 Therefore, the greatest S&P 500 index futures price at which we will receive a margin call is
930.88.
Question 5.14. An arbitrageur believing that the observed forward price, F(0,T), is too low will undertake a
reverse cash and carry arbitrage: Buy the forward, short sell the stock and lend out the proceeds
from the short sale. The relevant prices are therefore the bid price of the...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
 Spring '14
 NguyenThiMaiLan
 Derivatives

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