FIN 413 – RISK MANAGEMENT
Forward and
Futures Prices
Topics to be covered
•
•
•
•
•
•
Compounding frequency
Assumptions and notation
Forward prices
Futures prices
Cost of carry
Delivery options
Suggested questions from Hull
6th edition: #4.4, 4.10, 5.2, 5.5, 5.6, 5.14
5th edition: #4.4, 4.9, 5.2, 5.5, 5.6, 5.14
Compounding frequency
• Interest can be compounded with varying
frequencies.
• We will often assume that interest is
compounded continuously.
• Two rates of interest are said to be equivalent
if for any amount of money invested for any
length of time, the two rates lead to identical
future values.
Annual compounding
• The interest earned on an investment in any
one year is reinvested to earn additional
interest in succeeding years.
• R ≡ EAR, effective annual rate
A
A(1+R)n
0
n
A(1+R)-n
A
0
n
FV = A(1+R)n
PV = A(1+R)-n
Compounding m times per year
• The year is divided into m compounding
periods. Interest earned in any compounding
period is reinvested to earn additional interest
in succeeding periods.
• Rm ≡ the annual (or nominal) rate of interest
compounded m times per year
• Rm/m ≡ the effective rate of interest for each
mth of a year
Compounding m times per year
FV = A(1+Rm/m)mn
A(1+Rm /m)mn
A
0
n
PV = A(1+Rm/m)-mn
A(1+Rm /m)-mn
0
A
n
Continuous compounding
R∞ ≡ the annual rate of interest compounded continuously
FV = lim
A(1+Rm/m)mn
m→∞
= AeR∞n
PV = lim
A(1+Rm/m)-mn
m→∞
= Ae-R∞n
A
AeR∞n
0
n
Ae-R∞ n
0
A
n
Euler’s number
2<e<3
e = 2.71828183…
infinite decimal expansion
Conversion formulas
Rm 

$100  1 

m


Rm 


1 
m 

mn
 $100 e R n
m
 e R
m
R
R
Rm 

 ln  1 

m


Rm 

 m ln  1 

m 

Conversion formulas
Rm 

$100  1 

m


mn
 $100e R n
m
Rm 

R
1


e


m


Rm 

R m
1 
e
m 

Rm
R

 m e
m
 1

Natural log function
Natural Logarithm Function
2
1
0
ln(x)
Properties:
-∞<ln(x)<∞, for 0<x<∞
ln(x)<0, for 0<x<1
ln(1) = 0
ln(x)>0, for x>1
ln(ax) = ln(a) + ln(x)
ln(a/x) = ln(a) - ln(x)
ln(ax) = xln(a)
ln(ex) = xln(e) = x
-1 0
1
2
3
-2
-3
-4
-5
x
4
5
6
Exponential function
Exponential Function
20
15
exp(x)
Properties:
ex>0, for -∞<x<∞
0<ex<1, for x<0
e0 = 1
ex>1, for x>0
e-x = 1/ex
exey = ex+y
(ex)y = exy
eln(x) = x
10
5
0
-3
-2
-1
0
x
1
2
3
Short selling in the spot market
Involves selling securities that you do not own and buying them
back later.
When you initiate a short sale, your broker borrows the securities
from another client and sells them on your behalf in the spot
market. You receive the proceeds of the sale.
Through your broker, you must pay the client any income received
on the securities.
At some later stage, you must buy the securities, close your short
position, and return the securities to the client from whom you
borrowed.
Ignoring the income foregone, short selling yields a profit if the
price of the security falls.
Sell
Buy
Example
Suppose you short sell IBM stock for 90 days. The cash
flow are:
Action
Security
Cash
Day 0
Dividend
Ex-Day
Day 90
Borrow shares
-
Return shares
Sell shares
-
Purchase
shares
+S0
-D
-S90
Note: Short selling is the opposite of buying.
Analysis: forward prices
• Forward contracts are easier to analyze than futures
contracts.
• We begin our analysis with them.
• We will consider forward contracts on the following
underlying assets:
–
–
–
–
Assets that provide no income.
Assets that provide a known cash income.
Assets that provide a known yield.
Commodities
• Later we will consider futures contracts.
Assumptions
There are some market participants (such as large
financial institutions) that:
- pay no transactions costs (brokerage fees, bid-ask
spreads) when they trade.
- are subject to the same tax rate on all profits.
- can borrow or lend at the risk-free rate of interest.
- exploit arbitrage opportunities as they arise.
Note: The quality of any theory is a direct result of the
quality of the underlying assumptions. The
assumptions determine the degree to which the
theory matches reality.
Notation
T : the time (in years) until the delivery date of a forward contract
S (or S0): the current spot price of the asset underlying a forward
contract
K : the delivery price specified in a forward contract
F (or F0): the current forward price
f : the current value of a forward contract to the long
-f : the current value of a forward contract to the short
r : the risk-free interest rate (expressed as an annual, continuously
compounded rate) for an investment maturing in T years
Note: In practice, r is set equal to the LIBOR with a maturity of T
years.
LIBOR
• LIBOR: London Interbank Offer rate
• The rate at which large international banks
are willing to lend to other large international
banks for a specified period.
• The rate at which large international banks
fund most of their activities.
• A variable interest rate.
• A commercial lending rate, higher than
corresponding Treasury rates.
Analysis
• Objective: to derive formulas for F and f.
• We will use arbitrage pricing methods.
• Note: The basis of any arbitrage is to sell
what is relatively overvalued and to buy what
is relatively undervalued.
Forward contract: UA provides no
income
Examples: forward contracts on non-dividendpaying stocks and zero-coupon bonds.
Proposition: F = SerT, in the absence of arbitrage
opportunities
Note: F = SerT > S
Forward contract: UA provides no
income
Proposition: F = SerT, in the absence of arbitrage opportunities
Proof: Suppose F > SerT.
Arbitrage strategy (to be implemented today):
• Buy one unit of the UA in the spot market by borrowing S dollars for
T years at rate r.
• Short a forward contract on one unit if the UA.
At time T:
• Sell the UA for F dollars under the terms of the forward contract.
• Repay the bank SerT dollars.
Arbitrage profit per unit of UA = [F – SerT ] > 0.
S is bid up and F is bid down.
Forward contract: UA provides no
income
Suppose F < SerT.
Arbitrage strategy (to be implemented today):
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars.
Invest this for T years at rate r.
At
•
•
•
time T:
The proceeds from the sale/short sale have grown to SerT dollars.
Buy the UA for F dollars under the terms of the forward contract.
Return the UA to your portfolio or to the client from whom it was borrowed.
Arbitrage profit per unit of UA = [SerT – F ] > 0.
F is bid up and S is bid down.
Thus: F = SerT
Alternative derivation of formula
• Spot transaction
– Price agreed to.
– Price paid/received.
– Item exchanged.
• Prepaid forward contract
– Price agreed to.
– Price paid/received.
– Item exchanged in T years.
• Forward contract
– Price agreed to
– Price paid/received in T years.
– Item exchanged in T years.
Alternative derivation of formula
Underlying asset provides no income:
FP = S
Explanation: With a prepaid forward contract, as compared to a
spot transaction, physical exchange of the asset is delayed T
years. But since the asset, by assumption, pays no income to
the holder, the holder neither receives nor foregoes income due
to the delay.
F = FP erT = SerT
Explanation: The forward contract allows the long to delay
payment for T years and requires the short to delay receipt.
The long can earn interest on the cash that would otherwise
have been paid. The short foregoes this interest. The forward
price (which is arrived at by multiplying the prepaid forward
price, equal to S, by erT) compensates the short for the delay.
Forward contract: UA provides no
income
Proposition: f = S – Ke-rT
Proof:
In general: f = (F – K )e-rT
We derived: F = SerT
Thus: f = (SerT – K )e-rT = S – Ke-rT
Also: -f = -(F – K )e-rT = (K – F )e-rT = Ke-rT - S
Forward contract: UA provides no
income
We derived: f = S – Ke-rT
K
Thus: f > 0 iff S > Ke-rT
The value
today of the
UA in the spot
market.
0
The value today of
the price that the
long has agreed to
pay for the asset in
T years.
T
Forward contract: UA provides no
income
We derived: -f = Ke-rT – S
K
Thus: -f > 0 iff Ke-rT > S
The value today of
the price that the
short has agreed to
receive in T years
for the UA.
The value
today of the
UA in the spot
market.
0
T
Example: #5.9, page 121
T = 1 year
S = $40
r = 10%
0
(a) F = SerT = $40e(0.10×1) = $44.21
f = S – Ke-rT
= $40 – $44.21e-(0.10×1)
=0
1
Example (continued)
(b) T = ½ year
S = $45
r = 10%
0
F = S erT = $45e(0.10×0.5) = $47.31
f = S – Ke-rT
= $45 – $44.21e-(0.10×0.5)
= $2.95
0.5
1
Creating a forward contract
synthetically
A security is “created synthetically” by
assembling a portfolio of traded assets that
replicates the payoff to the security.
A long position in a forward contract can be
created synthetically by:
1. Buying the UA with borrowed funds.
2. Buying a call option and writing a put
option.
Creating a forward contract
synthetically
Method 1:
Consider a forward contract on a stock with a delivery
date in T years. The stock will pay no dividends
during the next T years.
The forward contract can be created synthetically by
buying the stock with borrowed funds.
r ≡ the annual, continuously compounded rate at which
funds can be borrowed.
S0 ≡ the current price of the stock.
Creating a forward contract
synthetically
Long position in
forward contract
Replicating
portfolio (the
stock and the
borrowed funds)
Cash flow at time 0
Zero
Zero
Cash outflow at time
K = F0 = S0erT
S0erT, to repay the
T
Security
Trader takes
possession of the
stock.
bank
Trader has full
possession of the
stock.
Creating a forward contract
synthetically
Value at time T of a long
position in a forward
contract = fT
= FT - K = ST – K
= ST – S0erT
Value at time T of
replicating portfolio:
Value of stock, ST
fT
ST
ST
-1 × what is owing to
the bank = -1 × S0 erT
Forward contract: UA provides a
known cash income
Examples: forward contracts on dividend-paying
stocks and coupon bonds.
I ≡ the present value of the income to be received
over the remaining life of the forward contract
Proposition: F = (S – I )erT, in the absence of
arbitrage opportunities
Forward contract: UA provides a
known cash income
Note: F = (S – I )erT < SerT
This price is lower than if the asset didn’t pay
income.
Forward contract: UA provides a
known cash income
Proposition: F = (S – I )erT, in the absence of arbitrage opportunities
Proof: Suppose F > (S – I )erT.
Arbitrage strategy (to be implemented today):
• Buy one unit of the UA in the spot market by borrowing S dollars for T
years at rate r.
• Short a forward contract on one unit if the UA.
Use the income from the asset to repay the loan.
At time T:
• Sell the UA for F dollars under the terms of the forward contract.
• Repay the bank (S – I )erT dollars.
Arbitrage profit per unit of UA = [F – (S – I )erT] > 0.
S is bid up and F is bid down.
Forward contract: UA provides a
known cash income
Suppose F < (S – I )erT.
Arbitrage strategy (to be implemented today):
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars.
Invest this for T years at rate r.
At
•
•
•
time T:
The proceeds from the sale/short sale have grown to (S – I )erT dollars.
Buy the UA for F dollars under the terms of the forward contract.
Return the UA to your portfolio or to the client from whom it was borrowed.
Arbitrage profit per unit of UA = [(S – I )erT – F] > 0.
F is bid up and S is bid down.
Thus: F = (S – I )erT
Alternative derivation of formula
• Spot transaction
– Price agreed to.
– Price paid/received.
– Item exchanged.
• Prepaid forward contract
– Price agreed to.
– Price paid/received.
– Item exchanged in T years.
• Forward contract
– Price agreed to
– Price paid/received in T years.
– Item exchanged in T years.
Alternative derivation of formula
Underlying asset provides a known cash income:
FP = S - I
Explanation: With a prepaid forward contract, as compared to a spot
transaction, physical exchange of the asset is delayed T years. As a
result of the delay, the long foregoes income with present value I and
the short receives this income. Thus, the price paid by the long and
received by the short is reduced by amount I.
F = FP erT = (S – I )erT
Explanation: The forward contract allows the long to delay payment for T
years and requires the short to delay receipt. The long can earn
interest on the cash that would otherwise have been paid. The short
foregoes this interest. The forward price (which is arrived at by
multiplying the prepaid forward price, equal to S - I, by erT)
compensates the short for the delay.
Forward contract: UA provides a
known cash income
Proposition: f = S – I – Ke-rT
Proof:
In general: f = (F – K )e-rT
We derived: F = (S – I )erT
Thus: f = [(S – I )erT – K]e-rT = (S – I )– Ke-rT
Also: -f = Ke-rT – (S – I )
Forward contract: UA provides a
known cash income
We derived: f = S – I – Ke-rT
K
Thus: f > 0 iff S > Ke-rT + I
The value
today of the
UA in the spot
market.
The value today of
the price that the
long has agreed to
pay for the asset in
T years.
0
T
The value today of
the income the long
foregoes as a result
of delaying
purchase of the
asset for T years.
Forward contract: UA provides a
known cash income
We derived: -f = Ke-rT – (S – I)
K
Thus: -f > 0 iff Ke-rT + I > S
The value today of
the price at which
the short has agreed
to sell the asset in T
years.
The value today of
the income the
short receives as a
result of delaying
sale of the asset for
T years.
0
T
The value
today of the
UA in the spot
market.
Example: #5.23, page 123
S = $50
r = 8%
T = 6/12
0
$1
$1
2/12
5/12
(a) I = $1e-(0.08×2/12) + $1e-(0.08×5/12) = $1.9540
F = (S – I )erT = (50 – 1.9540)e(0.08×6/12) = $50.0068
-f = -(S – I – Ke-rT)
= -(50 – 1.9540 – 50.0068e-(0.08×6/12)) = 0
6/12
Example (continued)
(b) S = $48
r = 8%
T = 3/12
$1
0
2/12
$1
3/12
5/12
I = $1e-(0.08×2/12) = $0.9868
F = (S – I)erT = (48 – 0.9868)e(0.08×3/12) = $47.9629
-f = -(S – I – Ke-rT)
= -(48 – 0.9868 – 50.0068e-(0.08×3/12)) = $2.00
6/12
Example (continued)
S = $50
T = 6/12
0
$1
$1
2/12
5/12
Term structure of interest rates:
Zero Rate (% per annum)
Maturity
7.80%
2 months
8.20%
5 months
(a) I = $1e-(0.078×2/12) + $1e-(0.082×5/12) = $1.9535
6/12
Forward contract: UA provides a
known yield
Examples: forward contracts on stock portfolios
and currencies.
q ≡ the average yield per annum expressed as a
continuously compounded rate
Proposition: F = Se(r-q)T, in the absence of
arbitrage opportunities
Forward contract: UA provides a
known yield
Note: F = Se(r-q)T < SerT
This price is lower than if the asset didn’t pay
income.
Forward contract: UA provides a
known yield
Proposition: F = Se(r-q)T, in the absence of arbitrage opportunities
Proof: Suppose F > Se(r-q)T.
Arbitrage strategy (to be implemented today):
• Buy one unit of the UA in the spot market by borrowing S dollars for T
years at rate r.
• Short a forward contract on one unit if the UA.
Use the income from the asset to repay the loan.
At time T:
• Sell the UA for F dollars under the terms of the forward contract.
• Repay the bank Se(r-q)T dollars.
Arbitrage profit per unit of UA = [F – Se(r-q)T ] > 0.
S is bid up and F is bid down.
Forward contract: UA provides a
known yield
Suppose F < Se(r-q)T.
Arbitrage strategy (to be implemented today):
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars.
Invest this for T years at rate r.
At
•
•
•
time T:
The proceeds from the sale/short sale have grown to Se(r-q)T dollars.
Buy the UA for F dollars under the terms of the forward contract.
Return the UA to your portfolio or to the client from whom it was borrowed.
Arbitrage profit per unit of UA = [Se(r-q)T – F ] > 0.
F is bid up and S is bid down.
Thus: F = Se(r-q)T
Alternative derivation of formula
• Spot transaction
– Price agreed to.
– Price paid/received.
– Item exchanged.
• Prepaid forward contract
– Price agreed to.
– Price paid/received.
– Item exchanged in T years.
• Forward contract
– Price agreed to
– Price paid/received in T years.
– Item exchanged in T years.
Alternative derivation of formula
Underlying asset provides a known yield:
FP = Se-qT
Explanation: FP equals the investment required in the asset today that will
yield one unit of the asset in T years when physical delivery occurs.
e-qT units of the asset will grow to e-qT × eqT = 1 unit of the asset in T
years, assuming that the income provided by the asset is reinvested in
the asset. e-qT units of the asset cost Se-qT today. Therefore,
FP = Se-qT .
F = FP erT = Se-qTerT = Se(r-q)T
Explanation: The forward contract allows the long to delay payment for T
years and requires the short to delay receipt. The long can earn
interest on the cash that would otherwise have been paid. The short
foregoes this interest. The forward price (which is arrived at by
multiplying the prepaid forward price, equal to Se-qT, by erT)
compensates the short for the delay.
Forward contract: UA provides a
known yield
Proposition: f = Se-qT – Ke-rT
Proof:
In general: f = (F – K )e-rT
We derived: F = Se (r-q)T
Thus: f = [Se(r-q)T – K ]e-rT = Se(r-q)T/erT – Ke-rT
= Se-qT – Ke-rT
Also: -f = Ke-rT – Se-qT
Example: #5.11, page 122
r = 9%
S = 300
T = 5/12
q = (5% + 2% + 2% + 5% + 2%)/5 = 3.2%
F = Se
(r-q)T
= 300e((0.09-0.032)×5/12) = 307.34
Forward prices & futures prices
• Like forward contracts, futures contracts are contracts for
deferred delivery.
• But, unlike forward contracts, futures contracts are marked to
market daily.
• Consider “corresponding” forward and futures contracts:
– Same underlying asset.
– Delivery date in two days.
• The contracts are identical except:
– Forward contract is settled at maturity.
– Futures contract is settled daily.
• Ignore taxes, transaction costs, and the treatment of margins.
• F ≡ the forward price
• G ≡ the futures price
Forward prices & futures prices
Forward price:
Payoff to buyer:
Futures price:
Payoff to buyer:
Day 0
Day 1
0
0
F0
F1
Day 0
Day 1
0
G1 – G0
G0
G1
Day 2
F2 = S2
F2 – K = S2 – F0
Day 2
G2 = S2
G2 – G1 = S2 – G1
Forward prices & futures prices
Example: Suppose we have:
Day 0: G0 = $2
Day 1: G1 = $1 with a 50% probability
= $3 with a 50% probability
Day 2: G2 = S2 since the futures contract
terminates.
Example (continued)
Suppose that the interest rate is a constant 10% (effective per
day).
On day 1, if G1 = $1: the futures buyer has a loss =
(G0 – G1) = $1. S/he would borrow this amount at
r = 10% and have to repay $1.10 on day 2.
On day 1, if G1 = $3: the futures buyer has a gain =
(G1 – G0) = $1. S/he would invest this amount at
r = 10% and have $1.10 on day 2.
Since there is a 50% chance of paying interest of $0.10 and a 50%
chance of earning interest of $0.10, there is no expected benefit
from marking to market on day 1.
Since the futures contract offers no benefit as compared to the
forward contract, G0 = F0.
Example (continued)
Now suppose that the interest rate is not constant. Suppose that
r = 12% on day 1 if G1 = $3 and r = 8% on day 1 if G1 = $1.
On day 1, if G1 = $1: the futures buyer has a loss =
(G0 – G1) = $1. S/he would borrow this amount at
r = 8% and have to repay $1.08 on day 2.
On day 1, if G1 = $3: the futures buyer has a gain =
(G1 – G0) = $1. S/he would invest this amount at
r = 12% and have $1.12 on day 2.
Now there is an expected benefit from marking to market =
(50% × $0.12 – 50% × $0.08) = $0.02.
Since the futures contract offers a benefit as compared to the
forward contract, G0 must exceed F0.
Example (continued)
Now suppose that the interest rate is not constant. Suppose that
r = 8% on day 1 if G1 = $3 and r = 12% at day 1 if G1 = $1.
On day 1, if G1 = $1: the futures buyer has a loss =
(G0 – G1) = $1. S/he would borrow this amount at
r = 12% and have to repay $1.12 on day 2.
On day 1, if G1 = $3: the futures buyer has a gain =
(G1 – G0) = $1. S/he would invest this amount at
r = 8% and have $1.08 on day 2.
Now the expected gain from marking of market =
(50% × $0.08 – 50% × $0.12) = -$0.02.
Since the forward contract offers a benefit as compared to the
futures contract, F0 must exceed G0.
Forward prices & futures prices
•
•
•
With this reasoning:
- G0 = F0 when interest rates are uncorrelated with the futures price.
- G0 > F0 when interest rates are positively correlated with the futures
price.
- F0 > G0 when interest rates are negatively correlated with the futures
price.
Empirical evidence:
- Differences between the forward and futures prices are usually trivial
once factors such as taxes, transaction costs, and the treatment of
margin are controlled for.
- Exceptions:
. Contracts on fixed income instruments, like T-bills. The prices of
T-bills are highly negatively correlated with interest rates. F0 > G0
. Long-lived contracts.
Formulas for F : use to calculate both forward prices and futures prices.
Stock index futures contracts
• Heavily traded. See National Post website.
• Stock index: a weighted average of the prices of a
selected number of stocks.
• Underlying: the portfolio of stocks comprising the
index.
• Examples of stock indices (futures exchanges):
– S&P/TSX Canada 60 Index (ME)
– S&P 500 Composite Index (CME)
– NYSE Composite Index (NYFE)
Stock index futures contracts
• A futures contract on an asset that provides
income.
• Formulas:
 F = Se(r-q)T
 F = (S – I )erT
 S denotes the current value of the index.
• Index arbitrage: what kind of trader might
engage in this arbitrage?
• F > Se(r-q)T
• F < Se(r-q)T
Stock index futures contracts
• Cash-settled contracts.
• More likely to lead to delivery.
• On the last trading day, the settlement price is set
equal to the closing value of the index.
• Multiplier (m):
– S&P 500 composite index futures, m = 250
– S&P/TSX Canada 60 index futures, m = 200
• The long gains if F2 > F1. The short gains if F2 < F1:
– F1: the futures price at the time the position is initiated.
– F2: the futures price at the time the position is terminated.
Example
On May 20, 2005, you go long two March 2006 futures contracts on
the S&P 500 Composite Index. The contract is trading at
1206.60. Suppose you hold the contract to expiration and the
index is at 1193.50 at that time. What is your gain/loss?
Solution:
F1 = 1206.60
F2 = 1193.50
Your loss = ((F1 – F2)×$250×2) = ((1206.60 – 1193.50)×$250×2) =
$6,550
Note:
1. If you had shorted the contracts, you would have gained $6,550.
2. If m = 1, your loss would have equaled $26.20.
Stock index futures contracts
•
•
•
•
•
S&P 500 composite index futures: m = 250
Mini S&P 500 futures: m = 50
Both of these contracts trade on CME.
See www.cme.com
Question: Who trades the mini? Designed for
individual investors, rather than professional
portfolio managers.
Forward and futures contracts on
currencies
See National Post website.
Foreign currency: a security that provides a known yield at rate q = rf
Our earlier formula, F = Se(r-q)T, becomes F = Se(r-rf )T
Notation:
r ≡ the domestic risk-free interest rate
rf ≡ the foreign risk-free interest rate
S ≡ the spot price of the foreign currency (or spot exchange rate)
expressed in units of the domestic currency, e.g., 1 CAD = 0.9270
USD
F ≡ the forward or futures price of the foreign currency expressed in
units of the domestic currency, e.g., 1 CAD = 0.9342 USD (1-year
forward)
Forward and futures contracts on
currencies
Proposition: F = Se(r-rf )T, in the absence of arbitrage opportunities
Proof: Suppose F > Se(r-rf )T.
Arbitrage strategy (to be implemented today):
• Buy one ₤ in the spot market by borrowing S dollars for T years at rate r.
• Short a forward contract on one ₤.
Use the income from the invested ₤ to repay the loan.
At time T:
• Sell the ₤ for F dollars under the terms of the forward contract.
• Repay the bank Se(r-rf )T dollars.
Arbitrage profit per ₤ = [F – Se(r-rf)T ] > 0.
S is bid up and F is bid down.
Forward and futures contracts on
currencies
Suppose F < Se(r-rf)T.
Arbitrage strategy (to be implemented today):
• Go long a forward contract on one ₤.
• Sell one ₤. This leads to a cash inflow of S dollars. Invest this for T years
at rate r.
At
•
•
•
time T:
The proceeds from the sale have grown to Se(r-rf )T dollars.
Buy one ₤ for F dollars under the terms of the forward contract.
Return the ₤ to your portfolio.
Arbitrage profit per ₤ = [Se(r-rf)T – F ] > 0.
F is bid up and S is bid down.
Thus: F = Se(r-rf )T
Forward contract: UA is a foreign
currency
For an asset that provides a known yield, we had:
f = Se-qT – Ke-rT
-f = Ke-rT – Se-qT
Foreign currency: a security that provides a known yield
at rate q = rf
Thus, for a forward contract on a foreign currency, we
have:
f = Se-rfT – Ke-rT
-f = Ke-rT – Se-rfT
Futures on commodities
Commodity: bulky, entails storage costs if held
Types:
1. Investment commodity: held primarily for
investment purposes, e.g., gold, silver
2. Consumption commodity: held primarily to
be used, e.g., oil, copper, canola
Investment commodities
Examples: gold, silver
Ignoring storage costs, these are assets that pay no
income. Thus: F = SerT.
But storage costs can be treated as negative income.
Letting U ≡ the present value of the storage costs
incurred during the life of a forward/futures
contract:
F = (S – I )erT = (S – (–U ))erT = (S + U )erT
Investment commodities
Proposition: F = (S + U )erT, in the absence of arbitrage opportunities
Proof: Suppose F > (S + U )erT.
Arbitrage strategy (to be implemented today):
• Buy one ounce of gold in the spot market, and arrange to store it,
by borrowing (S+U ) dollars for T years at rate r.
• Short a forward contract on one ounce of gold.
At time T:
• Sell the ounce for F dollars under the terms of the forward contract.
• Repay the bank (S+U )erT dollars.
Arbitrage profit per ounce = [F – (S+U )erT ] > 0.
S is bid up and F is bid down.
Investment commodities
Suppose F < (S+U )erT.
Arbitrage strategy (to be implemented today):
• Go long a forward contract on one ounce of gold.
• Sell one ounce of gold and forego storage costs. This leads to a cash
inflow of (S+U ) dollars. Invest this for T years at rate r.
At
•
•
•
time T:
The proceeds from the sale have grown to (S+U )erT dollars.
Buy one ounce for F dollars under the terms of the forward contract.
Return the ounce to your portfolio.
Arbitrage profit per ounce = [(S+U )erT – F ] > 0.
F is bid up and S is bid down.
Thus: F = (S+U )erT
Alternative derivation of formula
• Spot transaction
– Price agreed to.
– Price paid/received.
– Item exchanged.
• Prepaid forward contract
– Price agreed to.
– Price paid/received.
– Item exchanged in T years.
• Forward contract
– Price agreed to
– Price paid/received in T years.
– Item exchanged in T years.
Alternative derivation of formula
Underlying requires the payment of storage costs (expressed in present
value dollar terms):
FP = S + U
Explanation: With a prepaid forward contract, as compared to a spot
transaction, physical exchange of the asset is delayed T years. As a
result, the long forgoes storage costs with present value U and the
short has to pay these costs. Thus, the price paid by the long and
received by the short is increased by amount U.
F = FP erT = (S + U )erT
Explanation: The forward contract allows the long to delay payment for T
years and requires the short to delay receipt. The long can earn
interest on the cash that would otherwise have been paid. The short
foregoes this interest. The forward price (which is arrived at by
multiplying the prepaid forward price, equal to S + U, by erT)
compensates the short for the delay.
Investment commodities
As an alternative, storage costs can be expressed as a
proportion or percentage of the current spot price
of the commodity.
Storage costs can then be treated as a negative yield.
Letting u ≡ storage costs per annum as a proportion or
percentage of the spot price:
F = Se(r-q)T = Se(r-(-u))T = Se(r+u)T
Alternative derivation of formula
• Spot transaction
– Price agreed to.
– Price paid/received.
– Item exchanged.
• Prepaid forward contract
– Price agreed to.
– Price paid/received.
– Item exchanged in T years.
• Forward contract
– Price agreed to
– Price paid/received in T years.
– Item exchanged in T years.
Alternative derivation of formula
Underlying requires the payment of storage costs (expressed as a
percentage of the spot price):
FP = SeuT
Explanation: FP equals the investment required in the asset today that will
yield one unit of the asset in T years when physical delivery occurs.
euT units of the asset will grow to euT × e-uT = 1 unit of the asset in T
years, taking into consideration the storage costs that must be paid.
euT units of the asset cost SeuT. Therefore, FP = SeuT.
F = FP erT = SeuTerT = Se(r+u)T
Explanation: The forward contract allows the long to delay payment for T
years and requires the short to delay receipt. The long can earn
interest on the cash that would otherwise have been paid. The short
foregoes this interest. The forward price (which is arrived at by
multiplying the prepaid forward price, equal to SeuT, by erT)
compensates the short for the delay.
Consumption commodities
Examples: copper, oil, canola
Proposition: F ≤ (S + U )erT
F ≤ Se(r+u)T
F > (S + U )erT (F > Se(r+u)T)
Sell
Buy
Traders will respond. S will be
bid up and F will be bid down.
F < (S + U )erT (F < Se(r+u)T)
Buy
Sell
Traders may not respond. If
they don’t, S will not be bid
down and F will not be bid up.
Consumption commodities
Note: We can convert the inequalities to equalities
by using the concept of convenience yield: a
measure of the benefits of holding the physical
commodity.
Letting y ≡ the convenience yield, expressed as
an annual, continuously compounded rate:
F = (S + U )e(r-y )T
F = Se(r+u-y )T
Estimating convenience yield
Provide an estimate of the convenience yield of oil:
It is May 2007.
Current spot price (WTI) = $64.35
The August 2007 contract (NYMEX) is trading at $66.52.
Let u = 10%.
There are 3 months to maturity of the contract.
3-month LIBOR = 5.32%
F = Se(r+u-y)T
66.52 = 64.35e(0.0532+0.10-y)(3/12)
y = 2.0537%
Example: #5.15, page 122
S = $9
Storage costs = $0.24 per year payable quarterly in
advance
r = 10%
T = 9/12
$0.24/4
$0.24/4
$0.24/4
0
3/12
6/12
9/12
U = ($0.24/4) +($0.24/4)e-(0.10×3/12) + ($0.24/4)e(0.10×6/12)
= $0.1756
F = (S + U )erT = (9 + 0.1756)e(0.10×9/12) = $9.89
No-Arbitrage Bounds
The analysis has ignored transaction costs:
trading fees, bid-ask spreads, different
interest rates for borrowing and lending, and
the possibility that buying or selling in large
quantities will cause prices to change.
With transaction costs, there is not a single noarbitrage price but rather a no-arbitrage
region.
Example
A trader owns silver as part of a long-term investment portfolio.
There is a bid-offer spread in the market for silver. The trader
can buy silver for $12.02 per troy ounce and sell for $11.97 per
troy ounce. The six-month risk-free interest rate is 5.52% per
annum compounded continuously. For what range of six-month
forward prices of silver does the trader have an arbitrage
opportunity?
Solution: For silver:
F = (S + U )erT
F = Se(r+u)T
Assume U = u = 0 since we are given no information on storage
costs.
Thus, F = SerT in the absence of arbitrage opportunities.
Example (continued)
There is an arbitrage opportunity if:
1) F > SerT = $12.02e(0.0552×6/12) = $12.36
Sell
Buy
2) F < SerT = $11.97e(0.0552×6/12) = $12.31
Buy
Sell
The trader has an arbitrage opportunity for F > $12.36
and F < $12.31. There is no arbitrage opportunity
for $12.31 ≤ F ≤ $12.36.
Example (continued)
Now suppose that the trader must pay a $0.10
transaction fee per ounce of silver.
There is an arbitrage opportunity if:
1) F > SerT = ($12.02 + $0.10)e(0.0552×6/12) = $12.46
2) F < SerT = ($11.97 - $0.10)e(0.0552×6/12) = $12.20
The trader has an arbitrage opportunity for F > $12.46
and F < $12.20. There is no arbitrage opportunity
for $12.20 ≤ F ≤ $12.46.
Forward and futures contracts on
currencies
• If interest rates are expressed as annual rates
compounded continuously:
( r rf )T
F  Se
• If interest rates are expressed as equivalent
effective annual rates:
 1 r
F  S
 1 r
f

T



Cost of carry
Cost of carry (c): the cost of holding an asset, including
the interest paid to finance purchase of the asset
plus storage costs minus income earned on the asset.
c can be positive, zero, or negative.
The concept allows us to express our formulas for F in a
more general way:
• Investment asset: F = SecT
• Consumption asset: F = Se(c-y)T
IA: F = SecT
Cost of carry
Underlying asset
Security that
provides no income
Formula for F
F = SerT
CA: F = Se(c-y)T
Cost of carry
c=r
Portfolio underlying a F = Se(r-q)T
stock index
c=r–q
Foreign currency
F = Se(r-rf )T
c = r – rf
Investment
commodity
F = Se(r+u)T
c=r+u
Consumption
commodity
F = Se(r+u-y)T
c=r+u
Cost of carry
Investment asset: F = SecT
Consumption asset: F = Se(c-y)T
T = 0 implies F = Se0 = S
That is, the forward/futures price of an asset
equals its spot price at the time the contract
expires.
Exponential Function
20
Cost of carry
exp(x)
15
10
5
0
-3
-2
Investment asset: F = SecT
Consumption asset: F = Se(c-y)T
-1
0
1
2
x
∂F/∂S = the amount by which the forward (futures) price
changes in response to an infinitesimal change in the
spot price, ceteris paribus
Investment asset: ∂F/∂S = ecT > 0
Consumption asset: ∂F/∂S = e(c-y)T > 0
F and S are positively correlated.
3
Exponential Function
20
15
exp(x)
Cost of carry
10
5
0
-3
-2
-1
Investment asset: F = SecT
c > 0 implies ecT > 1 and
F>S
Normal, contango market
1
c < 0 implies 0 < ecT < 1
and F < S
Inverted market,
backwardation
Inverted Market
Normal Market
F
S
S
F
st trading
1st
1
trading
day
day
0
x
DP
1st trading
day
DP
2
3
Cost of carry
Consumptiom asset: F = Se
(c-y)T
c > y implies e (c-y)T > 1
and F > S
Normal, contango market
c < y implies 0 < e (c-y)T <
1 and F < S
Inverted market,
backwardation
Normal Market
Inverted Market
F
S
S
F
st trading
1st
1
trading
day
day
DP
1st trading
day
DP
Cost of carry
Investment asset: F = SecT
Consumption asset: F = Se(c-y)T
∂F/∂T = the amount by which the forward (futures) price changes in
response to an infinitesimal change in the time to expiration of
the contract, ceteris paribus
Investment asset: ∂F/∂T = SecT × c
c > 0 implies ∂F/∂T > 0: normal or contango market
c < 0 implies ∂F/∂T < 0: inverted market, backwardation
Consumption asset: ∂F/∂T = Se(c-y)T × (c – y)
c > y implies ∂F/∂T > 0: normal or contango market
c < y implies ∂F/∂T < 0: inverted market, backwardation
Normal market
•
•
•
•
c > 0 (investment asset)
c > y (consumption asset)
F>S
∂F/∂T > 0, that is, forward (futures) contracts
with longer times to expiration trade at higher
prices than forward (futures) contracts with
shorter times to expiration.
Inverted market
•
•
•
•
c < 0 (investment asset)
c < y (consumption asset)
F<S
∂F/∂T < 0, that is, forward (futures) contracts
with longer times to expiration trade at lower
prices than forward (futures) contracts with
shorter times to expiration.
Amaranth Advisors LLC
• The Connecticut-based hedge fund lost
about $6 billion (40% of its value) in
September 2006 trading natural gas
derivatives.
• G&M, September 22, 2006: “The problem
with oil and gas these days is that the
market is morphing from backwardation,
when spot prices are higher than prices for
delivery in the future, to contango, when
futures prices are higher than spot.”
Spread trades
• A spread trade provides exposure to the difference
between two prices.
• It is a long-short futures position.
• Example:
– Calendar spread: go long long-term contract and short
short-term contract on the same underlying asset, or vice
versa.
– Intercommodity spread: go long futures on commodity A
and short futures on commodity B
– Geographical spread: go long NYMEX oil futures and go
short London’s ICE Brent oil futures
• For speculators, it offers reduced risk.
Amaranth Advisors LLC
• Traders expected the market to be in backwardation but it has
moved into contango.
• They implemented spread trades based on this expectation.
Traders at Amaranth were
betting:
What has happened:
F
F
T
. Expectation of a cold winter,
active hurricane season,
instability in oil and gas
producing countries
T
.Winter of 2005-2006 was
warm, the hurricane season
was benign, supply of oil and
gas was relatively high
Delivery options for a futures
contract
• T ≡ the time to expiration of a forward or
futures contract.
• Forward contract: we know T.
• Futures contract: we must estimate T.
• Question: When during the delivery period of
a futures contract will the short choose to
make delivery?
Delivery options for a futures
contract
Normal Market:
Investment asset, c > 0
Consumption asset, c > y
0
DP
Short should deliver as
soon as possible.
T
0
DP
Delivery options for a futures
contract
Inverted Market:
Investment asset, c < 0
Consumption asset, c < y
0
DP
Short should deliver as
late as possible.
T
0
DP
Next class
• Hedging with futures