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BE332 Options and Futures
创建日期：2024-05-18 17:55:19
所属分类：经济Economics
标签： BE332
Options and Futures

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BE332 Options and Futures

Lecture 4: The cost of carry model (I)

2Lecture 4: The cost of carry model (I)
1. Short selling
2. Forward price vs. future price
3. Continuous compounding
4. Cost of carry model: pricing Financial Forward and Futures
5. Summary
Reading:
➢ Hull: Chapter 5
Learning outcomes
• Be able to use continuous compounding
• Understand short selling
• Be able to calculate forward and futures prices for various
kinds of contracts
• Be able to design arbitrage strategies when forward/futures
price is not in line with the theoretical price.
3
Short Selling
• Short selling involves selling securities you do not own
• Your broker borrows the securities from another client
and sells them in the market in the usual way
• At some stage you must buy the securities so they can
be replaced in the account of the client
• You must pay dividends and other benefits the owner
of the securities receives
• There may be a small fee for borrowing the securities
4
Example
• You short 100 shares when the price is $100 and close out the
short position three months later when the price is $90
• During the three months a dividend of $3 per share is paid
• What is your profit?
• What would be the loss if you had bought 100 shares?
5
Forward vs. Futures Prices
• When the short term interest rate is constant, the forward and
futures prices are equal.
• When interest rates vary unpredictably they are, in theory, slightly
different:
– A strong positive correlation between interest rates and the
asset price implies the futures price is slightly higher than the
forward price
– A strong negative correlation implies the reverse
• In most circumstances, the difference between the forward and
futures price is sufficiently small to be ignored.
6
Continuous compounding
• Consider an amount W invested for T years at an interest r per
annum. If interest is compounded annually, after T years the
original amount W will grow to
• If the amount is compounded m times per year, then after T
years the original amount W will grow to
• The limit as m tends to infinity is called continuous
compounding. Under continuous compounding it can be
shown that the amount W after T years grows to
• Discounting W at a continuously compounded rate r for T
years involves multiplying by

.)1( TrW +
.)1( mT
m
r
W +
. TrWe
7
.rTe−
8Cost of carry model is used to determine forward and futures prices.
Assumptions of the cost of carry model:
Perfect market:
a) No transaction costs
b) No taxes
c) No restrictions on short selling
d) Assets are perfectly divisible
e) Risk free borrowing and lending occur at the same rate
f) Everyone has the same information
Notation:
1) S0: Spot price today
2) F0: Forward or futures price today
3) T: Time until expiration
4) r: Continuously compounded risk-free interest rate per annum
Determination of Forward and futures Price
• Forward and futures are priced by imposing no-arbitrage
conditions.
• Because futures and forwards are written on both
commodities and financial assets, it is important that we
distinguish between investment assets and consumption
assets.
• An investment assets is an asset that is held for investment
purposes by a significant number of investors. For example,
stocks, bonds, gold and silver.
• A consumption asset is held primarily for consumption
purposes. For example, commodities such as copper, oil, and
pork bellies.
9
10
The cost is S0 and is certain to lead to a cash inflow of F0 at time T.
Therefore S0 must equal the present value of F0.
Spot
market
t=0
Buy one share of asset for S0
t=T
Sell the asset under the
terms of the forward
contract for the pre-
agreed forward price
F0
Cash inflow=F0
Cash outflow= S0
Forward
market
t=0
Enter into a short forward contract
to sell 1 share of the asset for F0 at
time T
Cash flow=0
Case1: investment asset paying no dividend
rTeF −= 0
rTeSF 00 =0S
• If , arbitrageurs will buy the asset in the spot market
and enter into a short position in the forward contract,
agreeing to sell the asset in the future at the agreed forward
price.
• If , arbitrageurs will short sell the asset and enter into
a long position in the forward contract, agreeing to buy the
asset in the future at the agreed forward price.
11
rTeSF 00 
rTeSF 00 
Arbitrage opportunities if or
Forward contract is overpriced relative to
the spot price.
Action now:
1. Borrow S0 at the risk free rate r for time T
2. Buy one unit of asset
3. Enter into forward contract to sell asset in
time T for F0
Forward contract is underpriced relative to
the spot price.
Action now:
1. Short one unit of asset to realize S0
2. Invest S0 at the risk free rate r for time T
3. Enter into forward contract to buy asset in
time T for F0
Action in time T:
1. Sell asset for F0 under the terms of the
forward contract (cash inflow)
2. Use S0e
rt to repay loan with interest
(cash outflow)
Action in time T:
1. Receive S0e
rt from investment (cash
inflow)
2. Buy asset for F0 under the terms of the
forward contract (cash outflow)
3. Close short position in the spot market
Profit realized: F0 - S0e
rt Profit realized: S0e
rt - F0 12
Tr
eSF 00 
Tr
eSF 00 
Tr
eSF 00 
Tr
eSF 00 
Example
13
1) Consider a stock that is priced at £40. It
pays no dividend. The risk free interest rate is
5% p.a. with continuous compounding. What is
the forward price for a 3-month stock forward
contract?
2) What trades would you perform if you
observe the forward price in the OTC market
is i) £44 or ii) £36, which is different from the
price you calculated in 1).
14
Case 2: Investment asset with known income
(present value I)
The cost is S0 and is certain to lead to a cash inflow of F0 at time T and
an income with a present value of I.
Spot market t=0
Buy one share of asset for S0
t=T
Sell the asset under the
terms of the forward
contract for the pre-agreed
forward price F0
Cash inflow= F0
Cash outflow= S0
Present value of the Income= I
Forward
market
t=0
Enter into a short forward contract to sell
1 share of the asset for F0 at time T
Cash flow= 0
rTeISF )( 00 −=rTeFI −+= 00S
15
Example
1) Consider a stock that is priced at £40. It pays a known
dividend of £5 after 6 months. The risk free interest rate is 5%
p.a for 6 months and 6% p.a for a year with continuous
compounding. What is the forward price for a one-year stock
forward contract?
2) What trades would you perform if you observe the forward
price in the OTC market is i) £44 or ii) £36, which is different
from the price you calculated in 1).
Answer
1) According to no arbitrage rule, the forward price should be
equal to:
F0 = (40-5×exp(-0.05×0.5)) ×exp(0.06) = 37.30
16
IS0
erT
17
2)
i). £44
If F0 =44, the forward contract is overpriced as it is higher than its
theoretical price 37.30; and the stock is relatively underpriced in
the spot market.
Therefore, an arbitrageur can do the following trading today:
➢ In the forward market, sell a forward contract
➢ In the spot market, borrow £40, buy one share of stock.
(Note: here we need to be clear about how much we need
to borrow for 6 months and how much for 1 year because
interest rates for different terms are different.)
rTeISF )( 00 −
By the time dividend is paid (6 months later):
➢ Receive dividend, pay off part of the loan
By the end of the forward contract (1 year later):
➢ Deliver the stock under the terms of the forward contract
at the agreed forward price: 44
➢ Pay off the remaining loan, harvest risk free profit.
Now let’s do the calculation for the detailed cash flows:
18
• So first we need to calculate the present value of the dividend:
5×e-0.05×0.5= £4.88, which can be used to repay part of the loan
once received. So of the £40 borrowed, £4.88 is borrowed for
6 months at 5% p.a.
• And the remaining £35.12 (i.e. 40 – 4.88) is borrowed for a
year at 6% p.a. The amount owing at the end of the year is
therefore: 35.12×exp(0.06×1) = 37.30
• After repaying the loan the arbitrageur would net a profit of
44 – 37.30 = £6.70
19
20
(ii) £36
If F0 =36, then the forward contract is underpriced as it is lower
than its theoretical price 37.30; and the stock is relatively
overpriced in the spot market.
Therefore, an arbitrageur should do the following trading today:
➢ In the forward market, buy a forward contract
➢ In the spot market, short sell the stock, invest proceeds in
the bank
By the time dividend needs to be paid (6 months later):
➢ Withdraw part of the money to payoff the dividend
By the end of the forward contract (1 year later)
➢ Withdraw the remaining money from the bank
➢ Take delivery of the share under the terms of the forward
contract for the agreed price F0, and use the share to
close out the short position in the spot market.
rTeISF )( 00 −
21
Here is the detailed calculation:
➢ Of the £40 received from shorting the stock, £4.88 (present value
of the dividend £5) is invested for 6 months at 5% p.a. so it grows
into an amount sufficient for paying the dividend.
➢ The remaining £35.12 (i.e. 40 – 4.88) is invested at 6% p.a. for a
year and grows to £37.30.
➢ Under the terms of the forward contract, £36 is paid for buying
the stock which is then returned to the brokerage firm who lent the
stock. The arbitrageur net a profit of :
37.30 – 36 = £1.30
• Arbitrage
Forward price =£44 Forward price =£36
Action now:
Borrow £40: £4.88 for 6 months and £35.12
for a year;
Buy 1 unit of asset;
Enter into a forward contract to sell the asset
in 1 year for £44.
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