THE AUSTRALIAN NATIONAL UNIVERSITY
RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES
AND STATISTICS
FINM3003/7003: CONTINUOUS TIME FINANCE
Tutorial and Practice Examples 2022, No. 1 Selected Solutions
1. Calculate the fair price of a nine-month future on an asset that pays no
income. The nine-month spot rate of interest is 10% p.a. and the current
spot price of the asset is $250.
Solution S0e
rT = 250e(0.10)(0.75) = 269.47.
2. Calculate the fair price of a 60-day future on an asset, the current spot
price of which is $105 and which will pay a single income payment of $8 in
45 days’ time. The 60-day spot rate of interest is 10% p.a. and the 45-day
spot rate is 7.5% p.a.
Solution 98.68255
3. Calculate the fair price of a 210-day future on an asset that pays a con-
tinuous dividend yield of 6% p.a. The 210-day spot rate of interest is 12%
p.a. and the current spot price of the asset is $230.
Solution 238.08
4. A forward contract is written on a non-dividend paying stock. The current
spot price of the asset is $65. The contract matures in 90 days and r(0, 90) =
4.00% p.a.
(i) Find the forward price. What is the value of the contract?
Solution 65.644, 0
(ii) Your client wants a 90-day forward contract with a delivery price set
at $60. What is the value of a long position in this contract?
Solution 5.5889
5. 3 months ago you entered into a forward contract to buy stock. Now, the
contract will mature in 100 days’ time and the delivery price is $50.25. Due
to changing circumstances you no longer want the contract. To offset it, you
enter into a new forward contract to sell stock. The current stock price is
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$45, the interest rate is 4.75% p.a., and no dividends will be paid over the
life of the contract.
(i) Find the forward price for the new contract.
Solution Ft = 45e
(0.0475)(100/365) = 45.5894.
(ii) What is the value of your net position when the two forward contracts
mature?
Solution −50.25 + 45.5849 = −4.66.
(iii) What is the present value today of your net position?
Solution −4.66e−(0.0475)(100/365) = −4.60.
6. A forward contract was written for an asset for delivery in 1 year with a
forward price of $45. Now, 6 months later, forward contracts with the same
maturity date are being written for a forward price of $50. Suppose r = 0.10.
(i) What is a long position in the original contract worth at this time?
(ii) Explain how the holder of the original contract can capture the value
calculated in (i).
Solution (i)
Vt = (Ft − F0)e−r(T−t) = 5e−(0.10)(0.5) = $4.756.
(ii) At Time t = 0.5 yr, holder should enter a short forward to deliver
asset. At Time 1 yr, pay over $45 under old contract, take delivery of asset,
then deliver it to fulfil new contract and receive $50. Left with $5 whose
present value is $4.76. To capture this value, borrow $4.76 at 10% now and
pay back $5 at Time 1 yr.
7. Suppose there is a futures contract on a portfolio of stocks that currently
are worth $100. The futures has a life of 90 days, and during that time the
stocks will pay dividends of $0.75 in 30 days, $0.85 in 60 days, and $0.90 in
90 days. The interest rate is 12% p.a.
(i) Find the price of the futures contract assuming no arbitrage opportu-
nities are present.
(ii) Find the value of the basis in dollars.
Solution
(i) Present value of dividends payable (at Time 0) is
D0 = 0.75e
−(0.12)(30/365) + 0.85e−(0.12)(60/365) + 0.90e−(0.12)(90/365) = 2.4498,
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so
f0 = (S0 −D0)erT = (100− 2.44980)e(0.12)(90/365) = 100.4800.
(ii) Basis = Spot Price - Futures Price = -0.4800.
8. (i) Price a one-year forward contract on gold. The current spot price is
$450 per ounce and the risk-free rate is rc = 7%. Suppose it costs $2 per
ounce to store gold for a year, the payment being made at the end of the
year. Show that
F0 = $484.6.
(ii) For the setup in (i), you observe that a market price for a forward
contract on one ounce is $500. Find the gain from an arbitrage strategy
which takes advantage of this mispricing, and show how to set up and work
the strategy.
Solution
Here S0 = $450, r = 0.07, cost of carry = $2 (payable at maturity).
(i) F0 = S0e
rT + c = 450e(0.07)(1) + 2 = $484.6287.
(ii) The quoted price of $500 is too high by comparison with the arbitrage
free price, and we would like to receive that extra amount. So at Time 0 enter
into a forward contract to deliver 1 oz gold in 1 year. Borrow $450 and buy
1 oz gold. At Time T = 1 year, deliver gold to fulfil contract and receive
$500. Pay back 450e0.07 on loan plus $2 storage charge. You’ll be left with a
risk-free profit of $15.37, at Time 1.
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