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Unformatted text preview: stock and the lending
interest rate. Also, she will incur the transaction costs twice. We have: 39 Part 2 Forwards, Futures, and Swaps Description
Long forward Today
0 In 9 months
S T − F0,T b
Sell short tailed position + S 0 e −δT
of the index
Pay twice transaction cost − 2 × k − ST b
Lend S 0 e −δT − 2 × k b
− S 0 e −δT + 2 × k TOTAL 0 (+ S
(+ S To avoid arbitrage, we must have ( ) (S b
0 ) b −δT
0 − 2 × k )× e r T b −δT
0 − 2 × k × e r T − F0,T e
e l ) l l − 2 × k × e r T − F0,T ≤ 0 . This is equivalent to l b
F0,T ≥ S 0 − 2 × k × e r T . Therefore, for any F0,T smaller than this bound, there exist arbitrage
opportunities. Question 5.16. a) The one-year futures price is determined as: F0 ,1 = 875e 0.0475 = 875 × 1.048646 = 917.57
One futures contract has the value of $250 × 875 = $218,750 . Therefore, the number of
contracts needed to cover the exposure of $800,000 is: $800,000 ÷ $218,750 = 3.65714 .
Furthermore, we need to adjust for the difference in beta. Since the beta of our portfolio exceeds
1, it moves more than the index in either direction. Therefore, we must increase the number of
contracts. The final hedge quantity is: 3.65714 × 1.1 = 4.02286 . Therefore, we should short-sell
4.02286 S&P 500 index future contracts.
As the correlation between the index and our portfolio is assumed to be one, we have no basis
risk and have perfectly hedged our position and transformed it into a riskless investment.
Therefore, we expect to earn the risk-free interest rate as a return over one year.
Question 5.18. The current exchange rate is 0.02E/Y, which implies 50Y/E. The euro continuously compounded
interest rate is 0.04, the yen continuously compounded interest rate 0.01. Time to expiration is
0.5 years. Plug the input variables into the formula to see that:
Yen/Euro forward = 0.02e (0.04−0.01)×0.5 = 0.02 × 1.015113 = 0.020302
= 50e (0.01−0.04 )×0.5 = 50 × 0.98511 = 49.2556 40 Chapter 5 Financial Forwards and Futures Question 5.20. a)
The Eurodollar futures price is 93.23. Therefore, we can use equation (5.20) of the main
text to back out the three-month LIBOR rate:
r91 = (100 - 93.23) × 1 1 91
= 0.017113 .
100 4 90 b)
We will have to repay principal plus interest on the loan that we are taking from the
following June to September. Because we shorted a Eurodollar futures, we are guaranteed the
interest rate we calculated in part a). Therefore, we have a repayment of:
$10,000,000 × (1 + r91 ) = $10,000,000 × 1.017113 = $10,171,130 41 Chapter 6
Commodity Forwards and Futures
The spot price of oil is $32.00 per barrel. With a continuously compounded annual risk-free rate
of 2%, we can again calculate the lease rate according to the formula:
1 F δ l = r − ln 0,T T S0 Time to expiration
12 months Forward Price Annualized lease
0.0999906 The lease rate is higher than the risk-free interest rate. The forward curve is downward sloping,
thus the prices of exercise 6.2. are an example of backwardation.
As we need to borrow a pencil to sell it short, we must pay the lender the lease rate for
the time we borrow the asset, i.e., until expiration of the contract in one year. After one year, we
have to pay back one pencil, which will cost us ST , the uncertain future pencil price, plus the
leasing costs: Total payment = S T + ST × (e 0.05 − 1) = S T e 0.05 = 1.05127 × S T .
It does not make sense to store pencils in equilibrium, because even if we have an active lease
market for pencils, the lease rate is smaller than the risk-free interest rate. Lending money at ten
percent is more profitable than lending pencils at five percent.
b) The equilibrium forward price is calculated according to our pricing formula: F0,T = S 0 × e (r −δ l )×T = $0.20 × e (0.10−0.05 )×1 = $0.20 × 1.05127 = $0.2103 ,
which is the price given in the exercise.
c) Let us first look at the different arbitrage strategies we can use in each case . c1) Pencils can be sold short. We can engage in our usual reverse cash and carry arbitrage: 42 Chapter 6 Commodity Forwards and Futures Transaction
Long forward Time 0
0 Time T=1
S T − F0,T Short-sell tailed pencil position, @ 0.05
proceeds @ 0.1
Total $0.19025 − ST -$0.19025 $0.2103 0 $0.2103 − F0,T For there to be no arbitrage, F0,T ≥ $0.2103
Suppose pencils cannot be sold short. Then we have no ability to create the short position
necessary to offset the pencil price risk from the long forward. Consequently, we are not able to
find a lower boundary for the pencil forward in this case.
c3) Pencils can be loaned. We engage in a cash and carry arbitrage:
Short forward Time 0
0 Time T=1
F0,T − S T Buy tailed pencil
position, lend @0.05
borrow @ 0.1
Total -$0.19025 ST $0.19025
F0,T − $0.2103 For there to be no arbitrage, F0,T ≤ $0.2103
c4) Suppose penci...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
- Spring '14