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BE332 Options and Futures
Lecture 4: The cost of carry model (I)
Presented by Dr. June Neo
1
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
• We will frequently use continuous compounding in this
module. http://www.picalc.com/
.)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.
Action now:
Short 1 unit of asset to realize £40
Invest £4.88 for 6 months and £35.12 for 1
year
Enter into a forward contract to buy the asset
in 1 year for £36
Action in 6 months:
Receive £5 dividend, which is used to repay
the first loan and interest.
Action in 6 months:
Receive £5 from 6-month investment, which
is returned to the brokerage firm who lent the
asset
Action in 1 year:
Sell the asset for £44 under the terms of the
forward contract;
Use £37.30 to repay the second loan and
interest.
Action in 1 year:
Receive £37.30 from 1-year investment
Buy asset for £36 under the terms of the
forward contract and close out the short
position
Profit realized: £6.7 (=44-37.30) Profit realized: £1.3(=£37.30-36)
22
23
Case 3. Investment asset with known average yield q
For example, a stock index would normally pay dividend with a known
yield rather than a cash amount. This means that the dividend (or income)
is expressed as a percentage of the asset’s price at the time the dividend is
paid (q) and is continuous. Define q as the average yield per annum on an
asset during the life of a forward contract with continuous compounding.
Spot
market
t=0
Buy one share of asset for S0
t=T
Sell eqT shares of the
asset under the terms of
the forward contract for
the pre-agreed forward
price F0
Cash inflow= eqT F0
Cash outflow= S0
Forward
market
t=0
Enter into a short forward contract to
sell eqT share of the asset for F0 at time
T
Cash flow=0
24
The costs is S0 and is certain to lead to a cash inflow of
eqT F0 at time T. Therefore S0 must equal the present value of
eqT F0.
Example 2
Consider a 3-month futures contract on the FTSE 100 index.
Suppose the dividend yield and risk free interest rate are 3% p.a. ad
5% p.a., respectively, with continuous compounding. The current
value of the index is 4000. What is the no-arbitrage forward price?
05.4020
4000 25.0)03.005.0(0
=
= −eF
rTqT eeF −= )( 0
TqreSF )(00
−=0S
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 eqT units
of assets at 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 eqT
units of assets at time T for F0
Action in time T:
1. Sell eqT units of 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 eqT units of assets for F0 under the
terms of the forward contract (cash outflow)
3. Close short position in the spot market
Profit realized: F0 e
qT- S0e
rt Profit realized: S0e
rt - F0e
qT
25
( )
0
r q T
oF S e
− ( )0
r q T
oF S e
−
( )
0
r q T
oF S e
− ( )0
r q T
oF S e
−
• NOTE: The above arbitrage strategies are based on the
assumption that all income from the asset is reinvested in the
asset.
• As dividend is expressed as a percentage of the asset’s price at
the time the dividend is paid, the cash amount of the dividend is
uncertain due to uncertain asset prices.
• One way to resolve this issue is to reinvest the dividend in the
asset so that we can calculate the number of shares of the asset at
the maturity.
26
Quiz
1. An investor shorts 100 shares when the share price is $50 and
closes out the position six months later when the share price
is $43. The shares pay a dividend of $3 per share during the
six months. How much does the investor gain? _ _ _ _ _ _
2. The spot price of an investment asset that provides no income
is $30 and the risk-free rate for all maturities (with
continuous compounding) is 10%. What, is the three-year
forward price? _ _ _ _ _ _
3. Repeat question 2 on the assumption that the asset provides
an income of $2 at the end of the first year and at the end of
the second year.
27
4. BUG’s stock price S is $50 today. It pays a dividend of $0.25 after
two months and $0.30 after five months. The continuously
compounded interest rate is 4 percent per year. If the six-month
forward price is $51, the arbitrage profit that you can make today by
trading one forward contract and other securities is:
a)0 b) $0.18 c) $0.41 d) $0.54 e) None of these answers are correct.
5. Assume that interest rates are constant. Given a risk-free rate of 6
percent, a dividend yield of 2 percent, and index level of 1100, then
the stock market index futures price with delivery in 3 months
is:______
28
6. The following is NOT an implication of the cost-of-carry relation for
valuing a stock index futures contract:
a) the futures price depends directly upon the level of the stock
market index
b) if the stocks in the index increase the level of dividend payments
over the life of the futures contract, the futures price will fall,
with everything else constant
c) if the level of interest rates increases, the futures price will
increase, with everything else constant
d) if the level of interest rates increases, the futures price will
decrease, with everything else constant
e) None of these answers are correct.
29
BE332 Options and Futures
Lecture 5: The cost of carry model (II)
Presented by Dr. June Neo
30
31
Lecture 5: Cost-of-carry model (II)
1. Pricing Commodity Forward and Futures
2. Forward and futures on currencies
3. Cost of carry
Reading:
1. Hull: Chapter 5
Learning outcomes
• Be able to calculate forward and futures price for
commodities and currencies
• Be able to explain the covered interest rate parity
• Be able to identify arbitrage opportunities
• Be able to design arbitrage trading strategies and calculate
the riskless profit, including detailed cash flows of each
trading step
• Be able to explain the concept “cost of carry”
32
33
Commodity forward and futures
• Two types of commodities: consumption and investment
• Investment commodities are assets held by significant number
of people purely for investment purposes (Examples: gold,
silver)
• Consumption commodities are assets held primarily for
consumption (Examples: copper, oil)
Investment commodities forward and futures
• For a financial asset (like a stock) that has an income (like
dividend) with a present value of I during the forward
contract, the forward price is:
F0 = (S0 –I)e
rT
• Storage costs can be treated as negative income. If U is the
present value of all storage costs for storing the commodity
during the life of a forward contract, it follows that:
F0 = (S0 + U)e
rT
34
• Likewise, if storage costs can be considered as proportional
to the price of the commodity, then it can be treated as a
negative yield.
Therefore F0 = S0 e
(r+u)T ,
where u denotes the storage costs p.a. as a proportion of the
spot price
35
Example:
• Consider a one year futures contract on gold. Suppose the
cost is $2 per ounce per year payable at the end of the year.
Assume the spot price is 900 and the risk free interest rate
is 4% for all maturities. Calculate the futures price.
• Answer:
The futures price is given by:
36
72.938
)2900(
U)e (S F
1*04.01*04.0
rT
00
=
+=
+=
− ee
Why is this exponent is negative?
Why is this exponent is positive?
• Present value of $1:
– Annual compounding:
– Continuous compounding:
37
T
T
r
r
PV −+=
+
= )1(1
)1(
1
rTePV −=1
Arbitrage trading
• Suppose that
– Borrow S0 +U at the risk free, buy the commodity and pay
the storage costs.
– Short one futures contract.
– Profit at maturity:
• Suppose that
– Sell the commodity save the storage costs. Invest at the risk
free rate.
– Long one futures contract.
– Profit at maturity:
0 ( )
rT
oF S U e− +
0 ( )
rT
oF S U e +
38
0( )
rT
oS U e F+ −
0 ( )
rT
oF S U e +
Pricing consumption commodities futures
• held primarily for consumption purposes.
• ownership of a physical commodity may provide benefits that
are not obtained by the holders of futures contracts. WHAT
benefits?
• These benefits are called convenience yield.
39
40
Convenience yield
• Definition: the benefits from ownership of an asset that are
not obtained by the holder of a long futures contract on the
asset.
• It reflects the market’s expectation concerning the future
availability of the commodity.
• The greater the possibility that shortage will occur, the
higher the convenience yield, at which the futures price will
be discounted. Vice versa.
41
• Allowing y to represent the convenience yield, the new pricing
formula would be:
• The higher the convenience yield, the lower the futures or
forward price, with everything else being constant.
• If we assume the market price is the theoretical price we can
use the formula to back out the convenience yield
TyureSF )(00
−+=
Which of the following is true
A.The convenience yield
is always positive or zero.
B.The convenience yield is
always positive for an
investment asset.
C.The convenience yield is
always negative for a
consumption asset.
D.The convenience yield
measures the average
return earned by holding
futures contracts.
42A. B. C. D.
25% 25%25%25%
Futures and Forwards on Currencies
• A foreign currency is analogous to a security providing
a yield
• The yield is the foreign risk-free interest rate
• It follows that if rf is the foreign risk-free interest rate
• This relationship is also called the covered interest
parity relation in international finance.
43
F S e
r r Tf
0 0=
−( )
44
Proof of the covered interest rate parity relationship:
Let’s consider a forward contract from the perspective of a US
investor.
• S0 : the spot price (in US $) of one unit of the foreign
currency
• F0 : the forward price (in US $) of one unit of the foreign
currency
• r: the risk free interest rate in the domestic market
• rf : the risk free interest rate in the foreign market.
45
Spot
market
t=0 (Today)
Buy one unit of foreign currency for S0,
Invest it at the foreign risk free interest
rate rf until time T
t=T
Sell units of the
foreign currency under the
terms of the forward
contract for the pre-agreed
forward price F0
Cash inflow= ∙F0
Cash outflow= S0
Forward
market
t=0 (Today)
Enter into a short forward contract to
sell units of foreign currency at
the forward price F0 at time T
Cash flow=0
Trfe
Trfe
Trfe
Today’s cash outflow S0 is certain to lead to a cash inflow of at time T.
Therefore S0 must equal the present value of 0
Fe
Tr f
0Fe
Tr f
Proof:
46
( )Trr
Tr
rT
TrrT f
f
f eSF
e
e
SFFeeS
−
=== 000000
:implies arbitrage No
The CIP implies:
1) Domestic and foreign investment must offer the same final
wealth (assuming perfect capital markets) when the foreign
investment is covered with a forward contract.
2) Hence it should not be possible to make a riskless profit by
borrowing at a risk-free rate of interest in a domestic
currency, switching the borrowed funds into another currency,
investing them there at a risk-free rate and locking in a
forward sale to guarantee the rate of exchange back to the
domestic currency.
47
If we rearrange the above pricing relationship, we get the following:
From this equation we can see:
1) The covered interest parity (CIP) postulates an equilibrium
relation between the spot and the forward FX market, and
between the domestic and foreign interest rate market.
2) If the domestic interest rate is higher than the foreign interest
rate then the forward rate must be higher than the spot rate, so
that the difference in interest rates is exactly offset by the
difference between the forward and spot rate.
Trr fe
S
F )(
0
0 −=
Trr fe
S
F )(
0
0 ln)ln(
−
= TrrSF f )(lnln 00 −=−
48
Arbitrage with FX forwards and futures
Question: Suppose that the 2-year interest rates in the UK and the US
are 1% and 3% p.a. respectively. The current spot exchange rate is
2.0671 USD per GBP. What is the 2-year forward exchange rate?
Note: The quote currency is the home currency. The base currency
after the “per” is the foreign currency.
1515.2
0671.2
0
2)01.003.0()(
00
=
== −
−
F
eeSF
Trr f
49
Arbitrage Case 1:
Suppose first that the two year forward exchange rate is 1.96, which
is less than the theoretical forward price 2.1515. Then the GBP in
the forward market is underpriced and the GBP in the spot market is
relatively overpriced. What trading strategy will an arbitrageur
follow?
Arbitrage strategy:
Today:
1) In the spot market, borrow 1 unit of GBP, sell it for USD at the
spot price S0 , earn interest at rate r;
2) In the forward market, buy a forward contract for 1units
GBP at the forward price F0
At the end of the forward contract:
1) The USD has grown to S0
2) Buy 1units GBP using USD at the price F0 under the terms
of the forward contract
3) Payoff the loan in GBP. Profit:
Trr feSF
)(
00
−
TrrT feFeS 00 −
50
Borrow £1
(£1×e0.01×2)×1.96
=$1.9996
Convert to $2.0671 ($2.0671×e0.03×2)
= $2.1949
Profit (in $): 2.1949 – 1.9996 =
$0.1953
UK market 1% (loan)
US market 3%
proceeds
Amount needed
to repay the
loan
51
❑ Borrow £1 at 1% p.a for two years, convert to $2.0671 and
invest the USD at 3%.
❑The $2.0671 invested at 3% p.a. over two years grows to:
❑ Enter into a forward contract to buy £1.0202 (= £1×e0.01×2) for
£1.0202×1.96 = $1.9996
❑Of this $2.1949, $1.9996 is converted to pounds and is used to
pay back the principal and interest on the original £1 loan. The
overall strategy therefore gives rise to a riskless profit of:
($2.0671×e0.03×2)= $2.1949
Profit (in $): 2.1949 – 1.9996 = $0.1953
52
Arbitrage Case 2:
Suppose that the two year forward exchange rate is 2.23, which is
more than the theoretical forward price 2.1515. Then the GBP in the
forward market is overpriced and the GBP in the spot market is
relatively underpriced. What trading strategy will an arbitrageur
follow?
Today:
1) In the spot market, borrow S0 units of USD at r, buy 1 unit of
GBP at the spot price S0 , invest at the rf
2) In the forward market, sell a forward contract for 1units
GBP at the forward price F0
By the end of the forward contract:
1) Sell 1units GBP at the price F0 under the terms of the
forward contract
2) Payoff the loan in USD. Profit:
rTTr eSeF f 00 −
Trr feSF
)(
00
−
53
Convert to
£1
(£1×e0.01×2) ×2.23 =
$2.2750
Borrow $2.0671 ($2.0671×e
0.03×
2) = $2.1949
Profit (in $): 2.2750-2.1949=0.0801
UK market 1%
US market 3% (loan)
54
❑ Borrow $2.0671 at 3% p.a. for 2 years, convert to £1, and invest the
GBP at 1%.
❑ Enter into a forward contract to sell 1×exp(0.01×2) units of GBP for
1×exp(0.01×2) × 2.23 = $2.2750
❑The amount needed to pay off the USD borrowings is:
❑Therefore, the risk-free profit is:
1949.2$0671.2 203.0 =e
0801.0$1949.2$2750.2$ =−
Cost of carry
• The relationship between futures prices and spot prices can
be summarized in terms of the cost of carry.
• This measures the storage cost plus the interest that is paid
to finance the asset less the income earned on the asset.
– For a non-dividend paying stock, the cost of carry is r
– For a stock index, it is r-q
– For a currency, it is r-rf
– For a commodity that provides income at rate q and
requires storage costs at rate u, it is r-q+u;
Quiz
56
1. An exchange rate is 0.7000 and the six-month domestic and foreign
risk-free interest rates are 5% and 7% (both expressed with
continuous compounding). What is the six-month forward rate?
Give four decimal places _ _ _ _ _ _
2. The two-month interest rates in Switzerland and the United States
are 2% and 5% per annum, respectively, with continuous
compounding. The spot price of the Swiss franc is $0.8000. The
futures price for a contract deliverable in two months is $0.8100.
What arbitrage opportunities does this create?
3. The current USD/euro exchange rate is 1.4000 dollar per euro. The
six month forward exchange rate is 1.3950. The six month USD
interest rate is 1% per annum continuously compounded. Estimate
the six month euro interest rate.
4. Which of the following is a consumption asset (circle one)
a)The S&P 500 index
b)The Canadian dollar
c)Copper
d)IBM shares
5. Alloyum costs $0.10 per month to store (which is paid up front) but gives a
convenience yield of $0.12 per month (which is received on the maturity date). If
Alloyum’s spot price S is $200 per ounce and the continuously compounded
interest rate r is 5 percent per year, then the six-month forward price is:_____?
6. COMIND index is computed by averaging commodity prices. Compute the six-
month forward price for this index if the spot price is 1,000 and the continuously
compounded annual rates for various costs and benefits are 5 percent for the
interest rate, 2 percent for the dividend yield, 3 percent for the storage cost, and 1
percent for the convenience yield.
Lecture 4: The cost of carry model (I)
Presented by Dr. June Neo
1
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
• We will frequently use continuous compounding in this
module. http://www.picalc.com/
.)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.
Action now:
Short 1 unit of asset to realize £40
Invest £4.88 for 6 months and £35.12 for 1
year
Enter into a forward contract to buy the asset
in 1 year for £36
Action in 6 months:
Receive £5 dividend, which is used to repay
the first loan and interest.
Action in 6 months:
Receive £5 from 6-month investment, which
is returned to the brokerage firm who lent the
asset
Action in 1 year:
Sell the asset for £44 under the terms of the
forward contract;
Use £37.30 to repay the second loan and
interest.
Action in 1 year:
Receive £37.30 from 1-year investment
Buy asset for £36 under the terms of the
forward contract and close out the short
position
Profit realized: £6.7 (=44-37.30) Profit realized: £1.3(=£37.30-36)
22
23
Case 3. Investment asset with known average yield q
For example, a stock index would normally pay dividend with a known
yield rather than a cash amount. This means that the dividend (or income)
is expressed as a percentage of the asset’s price at the time the dividend is
paid (q) and is continuous. Define q as the average yield per annum on an
asset during the life of a forward contract with continuous compounding.
Spot
market
t=0
Buy one share of asset for S0
t=T
Sell eqT shares of the
asset under the terms of
the forward contract for
the pre-agreed forward
price F0
Cash inflow= eqT F0
Cash outflow= S0
Forward
market
t=0
Enter into a short forward contract to
sell eqT share of the asset for F0 at time
T
Cash flow=0
24
The costs is S0 and is certain to lead to a cash inflow of
eqT F0 at time T. Therefore S0 must equal the present value of
eqT F0.
Example 2
Consider a 3-month futures contract on the FTSE 100 index.
Suppose the dividend yield and risk free interest rate are 3% p.a. ad
5% p.a., respectively, with continuous compounding. The current
value of the index is 4000. What is the no-arbitrage forward price?
05.4020
4000 25.0)03.005.0(0
=
= −eF
rTqT eeF −= )( 0
TqreSF )(00
−=0S
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 eqT units
of assets at 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 eqT
units of assets at time T for F0
Action in time T:
1. Sell eqT units of 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 eqT units of assets for F0 under the
terms of the forward contract (cash outflow)
3. Close short position in the spot market
Profit realized: F0 e
qT- S0e
rt Profit realized: S0e
rt - F0e
qT
25
( )
0
r q T
oF S e
− ( )0
r q T
oF S e
−
( )
0
r q T
oF S e
− ( )0
r q T
oF S e
−
• NOTE: The above arbitrage strategies are based on the
assumption that all income from the asset is reinvested in the
asset.
• As dividend is expressed as a percentage of the asset’s price at
the time the dividend is paid, the cash amount of the dividend is
uncertain due to uncertain asset prices.
• One way to resolve this issue is to reinvest the dividend in the
asset so that we can calculate the number of shares of the asset at
the maturity.
26
Quiz
1. An investor shorts 100 shares when the share price is $50 and
closes out the position six months later when the share price
is $43. The shares pay a dividend of $3 per share during the
six months. How much does the investor gain? _ _ _ _ _ _
2. The spot price of an investment asset that provides no income
is $30 and the risk-free rate for all maturities (with
continuous compounding) is 10%. What, is the three-year
forward price? _ _ _ _ _ _
3. Repeat question 2 on the assumption that the asset provides
an income of $2 at the end of the first year and at the end of
the second year.
27
4. BUG’s stock price S is $50 today. It pays a dividend of $0.25 after
two months and $0.30 after five months. The continuously
compounded interest rate is 4 percent per year. If the six-month
forward price is $51, the arbitrage profit that you can make today by
trading one forward contract and other securities is:
a)0 b) $0.18 c) $0.41 d) $0.54 e) None of these answers are correct.
5. Assume that interest rates are constant. Given a risk-free rate of 6
percent, a dividend yield of 2 percent, and index level of 1100, then
the stock market index futures price with delivery in 3 months
is:______
28
6. The following is NOT an implication of the cost-of-carry relation for
valuing a stock index futures contract:
a) the futures price depends directly upon the level of the stock
market index
b) if the stocks in the index increase the level of dividend payments
over the life of the futures contract, the futures price will fall,
with everything else constant
c) if the level of interest rates increases, the futures price will
increase, with everything else constant
d) if the level of interest rates increases, the futures price will
decrease, with everything else constant
e) None of these answers are correct.
29
BE332 Options and Futures
Lecture 5: The cost of carry model (II)
Presented by Dr. June Neo
30
31
Lecture 5: Cost-of-carry model (II)
1. Pricing Commodity Forward and Futures
2. Forward and futures on currencies
3. Cost of carry
Reading:
1. Hull: Chapter 5
Learning outcomes
• Be able to calculate forward and futures price for
commodities and currencies
• Be able to explain the covered interest rate parity
• Be able to identify arbitrage opportunities
• Be able to design arbitrage trading strategies and calculate
the riskless profit, including detailed cash flows of each
trading step
• Be able to explain the concept “cost of carry”
32
33
Commodity forward and futures
• Two types of commodities: consumption and investment
• Investment commodities are assets held by significant number
of people purely for investment purposes (Examples: gold,
silver)
• Consumption commodities are assets held primarily for
consumption (Examples: copper, oil)
Investment commodities forward and futures
• For a financial asset (like a stock) that has an income (like
dividend) with a present value of I during the forward
contract, the forward price is:
F0 = (S0 –I)e
rT
• Storage costs can be treated as negative income. If U is the
present value of all storage costs for storing the commodity
during the life of a forward contract, it follows that:
F0 = (S0 + U)e
rT
34
• Likewise, if storage costs can be considered as proportional
to the price of the commodity, then it can be treated as a
negative yield.
Therefore F0 = S0 e
(r+u)T ,
where u denotes the storage costs p.a. as a proportion of the
spot price
35
Example:
• Consider a one year futures contract on gold. Suppose the
cost is $2 per ounce per year payable at the end of the year.
Assume the spot price is 900 and the risk free interest rate
is 4% for all maturities. Calculate the futures price.
• Answer:
The futures price is given by:
36
72.938
)2900(
U)e (S F
1*04.01*04.0
rT
00
=
+=
+=
− ee
Why is this exponent is negative?
Why is this exponent is positive?
• Present value of $1:
– Annual compounding:
– Continuous compounding:
37
T
T
r
r
PV −+=
+
= )1(1
)1(
1
rTePV −=1
Arbitrage trading
• Suppose that
– Borrow S0 +U at the risk free, buy the commodity and pay
the storage costs.
– Short one futures contract.
– Profit at maturity:
• Suppose that
– Sell the commodity save the storage costs. Invest at the risk
free rate.
– Long one futures contract.
– Profit at maturity:
0 ( )
rT
oF S U e− +
0 ( )
rT
oF S U e +
38
0( )
rT
oS U e F+ −
0 ( )
rT
oF S U e +
Pricing consumption commodities futures
• held primarily for consumption purposes.
• ownership of a physical commodity may provide benefits that
are not obtained by the holders of futures contracts. WHAT
benefits?
• These benefits are called convenience yield.
39
40
Convenience yield
• Definition: the benefits from ownership of an asset that are
not obtained by the holder of a long futures contract on the
asset.
• It reflects the market’s expectation concerning the future
availability of the commodity.
• The greater the possibility that shortage will occur, the
higher the convenience yield, at which the futures price will
be discounted. Vice versa.
41
• Allowing y to represent the convenience yield, the new pricing
formula would be:
• The higher the convenience yield, the lower the futures or
forward price, with everything else being constant.
• If we assume the market price is the theoretical price we can
use the formula to back out the convenience yield
TyureSF )(00
−+=
Which of the following is true
A.The convenience yield
is always positive or zero.
B.The convenience yield is
always positive for an
investment asset.
C.The convenience yield is
always negative for a
consumption asset.
D.The convenience yield
measures the average
return earned by holding
futures contracts.
42A. B. C. D.
25% 25%25%25%
Futures and Forwards on Currencies
• A foreign currency is analogous to a security providing
a yield
• The yield is the foreign risk-free interest rate
• It follows that if rf is the foreign risk-free interest rate
• This relationship is also called the covered interest
parity relation in international finance.
43
F S e
r r Tf
0 0=
−( )
44
Proof of the covered interest rate parity relationship:
Let’s consider a forward contract from the perspective of a US
investor.
• S0 : the spot price (in US $) of one unit of the foreign
currency
• F0 : the forward price (in US $) of one unit of the foreign
currency
• r: the risk free interest rate in the domestic market
• rf : the risk free interest rate in the foreign market.
45
Spot
market
t=0 (Today)
Buy one unit of foreign currency for S0,
Invest it at the foreign risk free interest
rate rf until time T
t=T
Sell units of the
foreign currency under the
terms of the forward
contract for the pre-agreed
forward price F0
Cash inflow= ∙F0
Cash outflow= S0
Forward
market
t=0 (Today)
Enter into a short forward contract to
sell units of foreign currency at
the forward price F0 at time T
Cash flow=0
Trfe
Trfe
Trfe
Today’s cash outflow S0 is certain to lead to a cash inflow of at time T.
Therefore S0 must equal the present value of 0
Fe
Tr f
0Fe
Tr f
Proof:
46
( )Trr
Tr
rT
TrrT f
f
f eSF
e
e
SFFeeS
−
=== 000000
:implies arbitrage No
The CIP implies:
1) Domestic and foreign investment must offer the same final
wealth (assuming perfect capital markets) when the foreign
investment is covered with a forward contract.
2) Hence it should not be possible to make a riskless profit by
borrowing at a risk-free rate of interest in a domestic
currency, switching the borrowed funds into another currency,
investing them there at a risk-free rate and locking in a
forward sale to guarantee the rate of exchange back to the
domestic currency.
47
If we rearrange the above pricing relationship, we get the following:
From this equation we can see:
1) The covered interest parity (CIP) postulates an equilibrium
relation between the spot and the forward FX market, and
between the domestic and foreign interest rate market.
2) If the domestic interest rate is higher than the foreign interest
rate then the forward rate must be higher than the spot rate, so
that the difference in interest rates is exactly offset by the
difference between the forward and spot rate.
Trr fe
S
F )(
0
0 −=
Trr fe
S
F )(
0
0 ln)ln(
−
= TrrSF f )(lnln 00 −=−
48
Arbitrage with FX forwards and futures
Question: Suppose that the 2-year interest rates in the UK and the US
are 1% and 3% p.a. respectively. The current spot exchange rate is
2.0671 USD per GBP. What is the 2-year forward exchange rate?
Note: The quote currency is the home currency. The base currency
after the “per” is the foreign currency.
1515.2
0671.2
0
2)01.003.0()(
00
=
== −
−
F
eeSF
Trr f
49
Arbitrage Case 1:
Suppose first that the two year forward exchange rate is 1.96, which
is less than the theoretical forward price 2.1515. Then the GBP in
the forward market is underpriced and the GBP in the spot market is
relatively overpriced. What trading strategy will an arbitrageur
follow?
Arbitrage strategy:
Today:
1) In the spot market, borrow 1 unit of GBP, sell it for USD at the
spot price S0 , earn interest at rate r;
2) In the forward market, buy a forward contract for 1units
GBP at the forward price F0
At the end of the forward contract:
1) The USD has grown to S0
2) Buy 1units GBP using USD at the price F0 under the terms
of the forward contract
3) Payoff the loan in GBP. Profit:
Trr feSF
)(
00
−
TrrT feFeS 00 −
50
Borrow £1
(£1×e0.01×2)×1.96
=$1.9996
Convert to $2.0671 ($2.0671×e0.03×2)
= $2.1949
Profit (in $): 2.1949 – 1.9996 =
$0.1953
UK market 1% (loan)
US market 3%
proceeds
Amount needed
to repay the
loan
51
❑ Borrow £1 at 1% p.a for two years, convert to $2.0671 and
invest the USD at 3%.
❑The $2.0671 invested at 3% p.a. over two years grows to:
❑ Enter into a forward contract to buy £1.0202 (= £1×e0.01×2) for
£1.0202×1.96 = $1.9996
❑Of this $2.1949, $1.9996 is converted to pounds and is used to
pay back the principal and interest on the original £1 loan. The
overall strategy therefore gives rise to a riskless profit of:
($2.0671×e0.03×2)= $2.1949
Profit (in $): 2.1949 – 1.9996 = $0.1953
52
Arbitrage Case 2:
Suppose that the two year forward exchange rate is 2.23, which is
more than the theoretical forward price 2.1515. Then the GBP in the
forward market is overpriced and the GBP in the spot market is
relatively underpriced. What trading strategy will an arbitrageur
follow?
Today:
1) In the spot market, borrow S0 units of USD at r, buy 1 unit of
GBP at the spot price S0 , invest at the rf
2) In the forward market, sell a forward contract for 1units
GBP at the forward price F0
By the end of the forward contract:
1) Sell 1units GBP at the price F0 under the terms of the
forward contract
2) Payoff the loan in USD. Profit:
rTTr eSeF f 00 −
Trr feSF
)(
00
−
53
Convert to
£1
(£1×e0.01×2) ×2.23 =
$2.2750
Borrow $2.0671 ($2.0671×e
0.03×
2) = $2.1949
Profit (in $): 2.2750-2.1949=0.0801
UK market 1%
US market 3% (loan)
54
❑ Borrow $2.0671 at 3% p.a. for 2 years, convert to £1, and invest the
GBP at 1%.
❑ Enter into a forward contract to sell 1×exp(0.01×2) units of GBP for
1×exp(0.01×2) × 2.23 = $2.2750
❑The amount needed to pay off the USD borrowings is:
❑Therefore, the risk-free profit is:
1949.2$0671.2 203.0 =e
0801.0$1949.2$2750.2$ =−
Cost of carry
• The relationship between futures prices and spot prices can
be summarized in terms of the cost of carry.
• This measures the storage cost plus the interest that is paid
to finance the asset less the income earned on the asset.
– For a non-dividend paying stock, the cost of carry is r
– For a stock index, it is r-q
– For a currency, it is r-rf
– For a commodity that provides income at rate q and
requires storage costs at rate u, it is r-q+u;
Quiz
56
1. An exchange rate is 0.7000 and the six-month domestic and foreign
risk-free interest rates are 5% and 7% (both expressed with
continuous compounding). What is the six-month forward rate?
Give four decimal places _ _ _ _ _ _
2. The two-month interest rates in Switzerland and the United States
are 2% and 5% per annum, respectively, with continuous
compounding. The spot price of the Swiss franc is $0.8000. The
futures price for a contract deliverable in two months is $0.8100.
What arbitrage opportunities does this create?
3. The current USD/euro exchange rate is 1.4000 dollar per euro. The
six month forward exchange rate is 1.3950. The six month USD
interest rate is 1% per annum continuously compounded. Estimate
the six month euro interest rate.
4. Which of the following is a consumption asset (circle one)
a)The S&P 500 index
b)The Canadian dollar
c)Copper
d)IBM shares
5. Alloyum costs $0.10 per month to store (which is paid up front) but gives a
convenience yield of $0.12 per month (which is received on the maturity date). If
Alloyum’s spot price S is $200 per ounce and the continuously compounded
interest rate r is 5 percent per year, then the six-month forward price is:_____?
6. COMIND index is computed by averaging commodity prices. Compute the six-
month forward price for this index if the spot price is 1,000 and the continuously
compounded annual rates for various costs and benefits are 5 percent for the
interest rate, 2 percent for the dividend yield, 3 percent for the storage cost, and 1
percent for the convenience yield.
