Derivatives Markets
Leonidas Rompolis
Chapter 2: Pricing Forward and Futures
In this chapter we examine how forward prices and futures prices are related to spot
prices of the underlying asset. Forward contracts are easier to analyze than futures
contracts because there is no daily settlement – only a single payment at maturity.
Luckily it can be shown that the forward price and futures price of an asset are usually
very close when the maturities of the two contracts are the same.
In a first part of the chapter we derive important general results on the relationship
between forward (or futures) prices and spot prices. In a second part we employ these
relationships on stock indices and currencies.
2.1. Short-selling
When you buy something, we are said to have a long position in that thing. For
example, if we buy a stock, we pay cash and receive the stock. Some time later, we
sell the stock and receive cash. This transaction is lending, in the sense that we pay
money today and receive money back in the future. The rate of return we receive may
not be known in advance, but it is a kind of loan nonetheless.
The opposite of a long position is a short position. Short-selling a stock entails
borrowing shares and then selling them, receiving the cash. Some time later, we buy
back the stock, paying cash for it, and return it to the lender. A short-sale can be
viewed, then, as just a way of borrowing money. When you borrow money from a
bank, you receive money today and repay it later, paying a rate of interest set in
advance. This is also what happens with a short-sale, except that you don’t necessarily
know the rate you pay to borrow.
Suppose an investor instructs a broker to short-sell IBM stock for 90 days. The broker
will carry out the instructions by borrowing the shares from another client and selling
them in the market. Table 1 depicts the cash flows.
Table 1: Cash flows associated with short-selling a share of IBM
Day 0
Dividend paid day
Day 90
Action
Borrow shares
Return shares
Security
Sell shares
Buy shares
Cash
S0
-D
-S90
Observe that if the share pays dividends, the short-seller must in turn make dividend
payments to the share-lender.
The investor is required to maintain a margin account with the broker. The margin
account consists of cash or marketable securities deposited by the investor with the
broker to guarantee that the investor will not walk away from the short position if the
share price increases. An initial margin is required and if there are adverse movements
(i.e., increases) in the price of the asset that is being shorted, additional margin may
be required. The margin account does not represent a cost to the investor. This is
because interest is usually paid on the balance in margin accounts and, if the interest
rate offered is unacceptable, marketable securities such as Treasury bills can be used
to meet margin requirements.
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Leonidas Rompolis
2.2. Interest rates
An interest rate in a particular situation defines the amount of money a borrower
promises to pay the lender. The interest rate applicable in a situation depends on the
credit risk. This is the risk that there will be a default by the borrower of funds, so that
the interest and principal are not paid to the lender as promised. The higher the credit
risk, the higher the interest rate that is promised by the borrower.
Treasury rates are the rates an investor earns on Treasury bills and Treasury bonds.
These are the instruments used by a government to borrow in its currency. It is usually
assumed that there is no chance that a government will default on an obligation
denominated in its own currency. Treasury rates are therefore totally risk-free rates in
the sense that an investor who buys a Treasury bill or bond is certain that interest and
principal payments will be made as promised.
The London Interbank Offered Rate (LIBOR) is a reference rate based on the
interest rates at which banks offer to lend funds to other banks in the London money
market. Large banks and financial institutions quote LIBOR in all major currencies
for maturities up to 12 months. A bank must satisfy certain creditworthiness criteria in
order to receive deposits from another bank at LIBOR. Typically it must have a AA
credit rating. LIBOR rates are not totally risk-free. There is small chance that a AArated financial institution will default on a LIBOR loan. However, they are close to
risk-free. Derivative traders regard LIBOR rates as a better indicator of the “true”
risk-free rate than Treasury rates, because a number of tax and regulatory issues cause
Treasury rates to be artificially low.
Suppose that you invest $1 at an annual interest rate, denoted as r, compounded m
times a year the amount received by the end of t years is:
mt
r 

1  
 m
If the compounding periods converge to infinity the compounding is instantaneous or
continuous. In this case,
mt
r 

lim 1    ert ,
m 
 m
and the amount received at the end of t years is ert .
2.3. Forward price for an investment asset
An investment asset is an asset that is held for investment purposes by significant
numbers of investors. Stocks and bonds are clearly investment assets. Gold and silver
are also examples of investment assets.
The following notation will be used throughout the course:
T: time until delivery date in a forward or futures contract (in year)
S0: price of the underlying asset today
F0: forward or futures price today
r: risk-free rate per annum expressed with continuous compounding, for an investment
maturing in T years.
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Leonidas Rompolis
2.3.1. No-income underlying asset
The easiest forward contract to value is one written on an investment asset that
provides the holder with no income. Non-dividend paying stocks and zero-coupon
bonds are typical examples.
In the following example we will present the basic idea of calculating the forward
price under the absence of any arbitrage opportunities. That is, we cannot generate a
positive cash flow either today or in the future with no net investment of funds and
with no risk.
Example: Consider a long forward contract to purchase a non-dividend paying stock
in 3 months. Assume the current stock price is $40 and the 3-month risk-free rate is
5% per annum.
Suppose that the forward price is at $43. An arbitrageur can borrow $40 at the riskfree rate of 5% per annum, buy one share, and short a forward contract to sell one
share in 3 months. At the end of the three months, the arbitrageur delivers the share
and receives $43. The sum of money required to pay off the loan is
40  e0.053/12  $40.50
and his profit at the end of the 3-months period, which is generated bearing no risk, is:
43 – 40.50 = $2.50
Suppose next that the forward price is at $39. An arbitrageur can short one share,
invest the proceeds of the short sale at 5% per annum for 3 months, and take a long
position in a 3-month forward contract. The proceeds of the short sale grows to
40  e0.053/12  $40.50
after 3 months. At the end of the 3 months, the arbitrageur pays $39, takes delivery of
the share under the terms of the forward contract, and uses it to close out the short
position. A net profit of
40.50 – 39 = $1.50
is therefore made at the end of the 3 months.
Thus if the price of the forward contract is greater or smaller to $40.50 an arbitrage
opportunity exists. We therefore deduce that for there to be no arbitrage the forward
price must be exactly $40.50.
To generalize this example, we consider a forward contract on an investment asset
with price S0 that provides no income. If T denotes the time to maturity, r is the riskfree rate and F0 is the forward price, then
F0  S0 erT
(1)
If F0  S0 erT , arbitrageurs can buy the asset and short forward contracts on the asset. If
F0  S0e rT , they can short the asset and enter into long forward contracts on it. In our
example, S0 = $40, r = 5%, T = 3/12 = 0.25, so the equation (1) gives
F0  S0 e rT  40  e0.053/12  $40.50
which is in agreement with our earlier conclusion.1
1
If short sales are not possible for the investment asset all that we require in order to derive equation
(1) is to assume that there be a significant number of people who hold the asset purely for investment
and they can take advantage of arbitrage opportunities as they occur.
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Leonidas Rompolis
Remark: We can also derive equation (1) using present value. If someone buys one
unit of the asset and enter into a short forward contract to sell it for F0 a time T, this
costs S0 and leads to a certain cash inflow of F0 at time T. Therefore S0 must equal the
present value of F0 discounted at the risk-free rate, that is,
S0  e  rT F0  F0  S0 e rT
2.3.2. Known income underlying asset
In this section we consider a forward contract on an investment asset that will provide
an a priori known cash income to the holder. Examples are stocks paying known
dividends and coupon bonds. We follow the same approach as in the previous section.
We first look at a numerical example and then review the formal arguments.
Example: Consider a long forward contract to purchase a coupon bond whose current
price is $900. The forward contract matures in 9 months. We also suppose that a
coupon payment of $40 is expected after 4 months. We assume that the 4-month and
9-month risk free rates are, respectively, 3% and 4% per annum.
Suppose that the forward price is at $910. An arbitrageur can borrow $900 to buy the
bond and short a forward contract. He knows that he will have a certain cash flow of
$40 after 4 months and a certain cash flow of $910 after 9 months. Thus a part of the
$900 can be borrowed at 3% and repaid after 4 months and the rest can be borrowed
at 4% and repaid after 9 months. Actually he can borrow
40  e0.034 /12  $39.60
at a 3% for 4 months and the remaining $860.40 is borrowed at 4% for 9 months. The
amount owning at the end of the 9-month period is
860.40  e0.049 /12  $886.60
His net profit, realized bearing no risk, is:
910 – 886.60 = $23.40
Suppose next that the forward price is at $870. An investor can short the bond and
enter into a long forward contract. Of the $900 realized from shorting the bond, the
$39.60 is invested for 4 months at 3% so that it grows into an amount of $40 in order
to pay the coupon. The remaining $860.40 is invested for 9 months at 4% and grows
to $886.60. Under the terms of the forward contracts $870 is paid to buy the bond and
the short position is closed out. The investor therefore gains
886.60 – 870 = $16.60
Like before it follows that if there are no arbitrage opportunities then the forward
price must be $886.60.
We can generalize from this example to argue that, when an investment asset will
provide a certain income I during the life of a forward contract, say at time T1, with
T1 < T, then we have

F0  S0  I  e
 rT1 T1
e
rT T
(2)
where rT1 is the risk-free rate for an investment maturing at T1 years and rT is the riskfree rate for an investment maturing at T years. The quantity I  e
value of the income.
 rT1 T1
is the present
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Leonidas Rompolis
Remark: We can also derive equation (2) using present value. If someone buys one
unit of the asset and enters into a short forward contract to sell it for F0 at time T, then
this costs S0 and is certain to lead to a cash inflow of I at time T1 and a cash inflow of
F0 at time T. Therefore S0 must equal the present value of I and F0 discounted at the
appropriate risk-free rate, that is,
S0  I  e
 rT1 T1

 F0  e  rT T  F0  S0  I  e
 rT1 T1
e
rT T
For stock indexes containing many stocks, it is common to model the dividend as
being paid continuously at a rate that is proportional to the level of the index; i.e., the
dividend yield (the annualized dividend payment divided by the stock price) is
constant. This is an approximation, but in a large stock index there can be dividend
payments on a large proportion of days. To model a continuous dividend, suppose that
the current index price is S0 and the annualized daily compounded dividend yield is q.
Then the dollar dividend over one day is:
q
D1 
S0
365
Now suppose that we reinvest dividends in the index. Because of the reinvestment,
after T years we will have more shares than we started with. Using continuous
compounding to approximate daily compounding we get
365T
q 

 eqT
Number of shares   1 

 365 
At the end of T years we have approximately eqT times the shares we had initially
(see Appendix for the derivation).
Suppose we wish to invest today in order to have one share at time T. We can buy
e  qT shares today. Because of dividend reinvestment, at time T, we will have eqT more
shares than we started with, so we end up with exactly one share. Since an investment
of e  qTS0 gives us one share at time T, the price of the forward contract to purchase
one share after time T is:
F0  S0 e  qT e rT  S0 e(r q)T
(3)
2.4. Valuing forward contracts
The value of a forward contract at the time it is first entered into is zero. At a later
stage it may prove to have positive or negative value. It is important for banks and
other financial institutions to value the contract each day, a procedure already referred
to as marking to market the contract.
Let K denote the delivery price for a contract that was negotiated at time 0, Ft the
forward price that would be applicable if we negotiated the contract at time t and Vt
the value of the forward contract at time t, prior to maturity. At time 0 (that is at the
beginning of the life of the forward contract) the delivery price K is set equal to the
forward price F0, and the value of the contract V0 is zero. As time passes, K stays the
same (as it is part of the definition of the contract) but the forward price changes and
the value of the contract becomes either positive or negative.
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Consider a long forward contract that has a delivery price of K which is held into your
portfolio and its value at time t is equal to Vt. Suppose that at time t you take a short
position at a forward contract with delivery price Ft. Of course the value of this
contract at time t is zero, therefore the value of the portfolio remains the same. At
time T you buy the underlying asset paying K (from the long position) and you sell it
for Ft (from the short position). Therefore you obtain a certain cash flow: Ft – K.
Under the absence of arbitrage the present value of this cash flow should be equal to
the value of the portfolio at time t, that is,
Vt   Ft  K  e  r (T  t )
(4)
Similarly, the value of a short forward contract with delivery price K is:
Vt   K  Ft  e  r (T  t )
(5)
If the investment asset provides no income equations (1) and (4) imply that:
Vt  St  Ke  r (T  t )
If the investment asset provides a known income I equations (2) and (4) imply that:
 r (T  t )
Vt  St  I  e T1 1  Ke  r (T  t )
Finally using equation (3) in conjunction with equation (4) gives the following
expression for the value of a long forward contract on an investment asset that
provides a yield q:
Vt  St e  q(T  t )  Ke  r(T  t )
2.5. Synthetic forward contracts
A market-maker or arbitrageur must be able to offset the risk of a forward contract. It
is possible to do this by creating a synthetic forward contract to offset a position in
the actual forward contract.
We assume that dividends are continuous and paid at the yield q, and hence equation
(3) is the appropriate forward price. We can then create a synthetic long portfolio
contract by buying the stock and borrowing to fund the position. To see how the
synthetic position works, recall that the payoff at expiration for a long forward
contract is
ST – F0
In order to obtain this same payoff, we invest e  qTS0 in the stock. This gives us one
share at time T. We borrow this amount so that we are not required to pay anything
additional at time 0. At time T we must repay S0 e(r q)T and we sell the stock for ST.
Thus at time T the total payoff is:
ST  S0 e(r q)T  ST  F0
This demonstrates that borrowing to buy the stock replicates the expiration payoff to a
forward contract.
Just as we can use the stock and borrowing to synthetically create a forward, we can
also use the forward to create synthetic stocks and bonds. Table 2 demonstrates that
we can go long a forward contract and lend the present value of the forward price to
synthetically create the stock.
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Transaction
Long one forward
Lend e  qTS0
Total
Leonidas Rompolis
Table 2: Synthetic share
Cash flows
Time 0
0
 qT
e S0
e  qTS0
Time T
ST – F0
S0 e(r q)T
ST
Table 3 demonstrates that if we can buy the stock and short the forward, we create
cash flows like those of a risk-free bond. The rate of return on this synthetic bond is
called the implied repo rate.
Transaction
Buy e  qT of the stock
Short one forward
Total
Table 3: Synthetic bond
Cash flows
Time 0
e  qTS0
0
 qT
e S0  e  rT F0
Time T
ST
F0 – ST
F0
To summarize we have shown that
Forward = Stock – zero-coupon bond
By rearranging this equation we derive the synthetic equivalents of Table 2 and 3:
Stock = Forward + zero-coupon bond
and
Zero-coupon bond = Stock – Forward
All these synthetic positions can be reversed to create synthetic short positions.
Now we will see how market-makers and arbitrageurs use these strategies. Suppose
an investor wishes to enter into a long forward position. The market-maker, as the
counterparty, is left holding a short forward position. He can offset this risk by
creating a synthetic long forward position. This can be constructed by borrowing to
buy the underlying asset as we have described earlier. The total cash flow at time T is
zero. Similarly, suppose the market-maker wishes to hedge a long forward position.
Then it is possible to reverse the above strategy (by shorting the underlying asset and
lend the cash received) to create a synthetic short forward position.
A strategy in which you offset the short forward position by creating a synthetic long
forward position is called a cash-and-carry. A cash-and-carry has no risk: You have
the obligation to deliver the asset but also own the asset. A reverse cash-and-carry
entails offsetting the long forward position by a synthetic short one.
We motivated the cash-and-carry as risk management by a market-maker. However,
an arbitrageur might also engage in a cash-and-carry. If the forward price is too high
relative to the stock price (i.e., F0  S0 e(r q)T ) then an arbitrageur can use the above
strategy to make a risk-free profit. If F0  S0 e(r q)T , the arbitrageur will select a reverse
cash-and-carry transaction.
Similarly, by comparing the implied repo rate with our borrowing rate, we have a
simple measure of whether there is an arbitrage opportunity. If the implied repo rate
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Leonidas Rompolis
exceeds the borrowing rate then there is an arbitrage opportunity. On the other hand,
if the borrowing rate exceeds the implied repo rate, there is no arbitrage opportunity.
2.6. Forward vs. futures prices
One should expect forward and futures prices to differ. This is due to the daily
settlement of gains/losses (marking-to-market procedure) which are then being
compounded at the risk-free rate. However, the appendix at the end of Chapter 5 of
the book (see page 126) provides an arbitrage argument to show that when risk-free
rate is constant and the same for all maturities, the forward price for a contract with a
certain delivery date is in theory the same as the futures price for a contract with that
delivery date. When interest rates vary unpredictably (as they do in the real world),
forward and futures prices are in theory no longer the same.
Suppose that futures prices are positively correlated with interest rates. When the
futures price increases an investor who holds a long futures contract makes an
immediate gain because of the daily settlement procedure. The positive correlation
indicates that interest rates have also increased. The gain will therefore tend to be
invested at a higher than average rate of interest. Similarly, when the futures prices
decreases, the investor will incur an immediate loss. This will tend to be financed at a
lower interest rate. An investor holding a forward contract rather than a futures
contract is not affected in this way by interest rate movements. It follows that a long
futures contract will be slightly more attractive than a similar forward contract, thus
futures prices will exceed forward ones. The investor who is long futures buys at a
higher price to offset the advantage of marking-to-market. When the forward prices
are negatively correlated with the interest rates, the futures prices will be less than an
otherwise identical forward price. The investor who is long futures buys at a lower
price to offset the disadvantage of marking-to-market.
As an empirical matter, forward and futures prices are very similar. The theoretical
difference arises from uncertainty about the interest on mark-to-market proceeds. For
short-lived contracts, the effect is generally small. However, for long-lived contracts,
the difference can be significant, especially for long-lived interest rates futures, for
which there is sure correlation between the interest rate and the price of the
underlying asset. For the rest of the course we will assume that forward and futures
prices are the same, both represented by the symbol F0.
2.7. Futures prices of stock indices
A stock index tracks changes in the value of a hypothetical portfolio of stocks. The
weight of the stock in the portfolio equals the proportion of the portfolio invested in
the stock. The percentage increase in the stock index over a small interval of time is
set equal to the percentage increase in the value of the hypothetical portfolio. A stock
index can be regarded as the price of an investment asset that pays dividends. The
investment asset is the portfolio of stocks underlying the index, and the dividends paid
by the investment asset are the dividends that would be received by the holder of the
portfolio. It is usually assumed that the dividends provide a known yield rather than a
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known cash income. If q is the dividend yield rate, equation (3) gives the futures price
F0 as:
F0  S0 e(r q)T
(6)
Example: Consider a 3-month futures contract on the S&P 500. Suppose that the
stocks underlying the index provide a dividend yield of 1% per annum, that the
current value of the index is 1,300, and that the continuously compounded risk-free
rate is 5% per annum. In this case, r = 0.05, S0 = 1,300, T = 0.25 and q = 0.01. Hence,
the futures price is given by:
F0  1,300  e(0.050.01)0.25  $1,313.07
In practice, the dividend yield on the portfolio underlying an index varies week by
week throughout the year. For example, a large proportion of the dividends on the
NYSE stocks are paid in the first week of February, May, August and November each
year. The chosen value of q should represent the average annualized dividend yield
during the life of the contract. The dividends used for estimating q should be those for
which the ex-dividend date is during the life of the futures contract.
The Chicago Mercantile Exchange (CME) trades a Nikkei 225 futures contract.
Compared to the S&P 500 index futures contract there is one very important
difference: Settlement of the contract is in a different currency (dollars) than the
currency for the index (yen). To see why this is important, consider a dollar-based
investor wishing to invest in the Nikkei 225 cash index. This investor must undertake
two transactions: changing dollars to yen and using yen to buy the index. When the
position is sold, the investor reverses these transactions, selling the index and
converting yen back to dollars. There are two sources of risk in this transaction: the
risk of the index, denominated in yen, and the risk that the yen/dollar exchange rate
will change. The Nikkei 225 futures contract is however denominated in dollars rather
than yen. Consequently, the contract insulates investors from currency risk, permitting
them to speculate solely on whether the index rises or falls. This type of contract is
called a quanto. Quanto contracts allow investors in one country to invest in a
different country without exchange rate risk.
2.8. Forward and futures contracts on currencies
We now move to consider forward and futures foreign currency contracts from the
perspective of a US investor. The underlying asset is one unit of the foreign currency.
We will therefore define the variables S0 as the current spot price in dollars of one
unit of the foreign currency and F0 the forward or futures price in dollars of one unit
of the foreign currency. However, it does not necessarily correspond to the way spot
and forward exchange rates are quoted. For major exchange rate other than the GBP,
the Euro, Australian dollar and New Zealand dollar, a spot or forward exchange rate is
normally quoted as the number of units of the currency that are equivalent to one US
dollar.
A foreign currency has the property that the holder of the currency can earn interest at
the risk-free interest rate prevailing in the foreign country. We define rf as the value of
the foreign risk-free interest rate when money is invested for time T. The variable r is
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the US dollar risk-free interest rate when money is invested for the same period of
time.
Consider an investor with m units of the foreign currency. There are two ways it can
be converted to dollars at time T. One is by investing it for T years at rf and entering
into a forward contract to sell the proceeds for dollars at time T. This generates
m  F0 e rf T dollars. The other is by exchanging the foreign currency for dollars in the
spot market and investing the proceeds for T years at rate r. This generates m  S0e rT
dollars. In the absence of arbitrage opportunities, the two strategies must give the
same result. Hence,
m  F0 e rf T  m  S0 e rT
so that
F0  S0 e(r  rf )T
(7)
Example: Suppose that the 2-year interest rates in Australia and the US are 5% and
7% respectively, and the spot exchange rate between Australian dollar (AUD) and
USD is 0.62 USD per AUD. From equation (7) the 2-year forward exchange rate
should be:
F0  0.62  e(0.07 0.05)2  0.6453
Suppose first that the 2-year forward exchange rate is less than this, say 0.63. An
arbitrageur can:
1. Borrow one unit AUD at 5% for 2 years, convert to 0.62 USD and invest the USD
at 7%
2. Enter into a forward contract to buy e0.052  1.1051 AUD for 1.1051 μ 0.63 =
0.6962.
The 0.62 USD that are invested at 7% grow to 0.62  e0.072  0.7131 USD in 2 years.
Of this, 0.6962 are used to purchase 1.1051 AUD under the terms of the forward
contracts. This is exactly enough to repay principal and interest on the one unit AUD
that are borrowed. The strategy therefore gives rise to a riskless profit of 0.7131 –
0.6962 = 0.0169.
Suppose next that the 2-year forward rate is greater than 0.6453, say 0.66. An
arbitrageur can:
1. Borrow one unit USD at 7% for 2 years, convert to 1/0.62 = 1.612 AUD and
invest the AUD at 5%.
2. Enter into a forward contract to sell 1.621 e0.052  1.782 AUD for 1.782 μ 0.66 =
1.176 USD.
The 1.612 AUD that are invested at 5% grow to 1.782 AUD in 2 years. The forward
contract has the effect of converting this to 1.176 USD. The amount needed to payoff
the USD borrowing is e0.072  1.150 USD. The strategy therefore gives rise to a
riskless profit of 1.176 – 1.150 = 0.0262 USD.
Equation (7) is identical to equation (3) with q replaced by rf. This is not a
coincidence. A foreign currency can be regarded as an investment asset paying a
known yield. The yield is the risk-free rate in the foreign currency. Suppose that the
interest rate on GBP is 5% per annum. To a US investor the GBP provides an income
equal to 5% of the value of the GBP. In other words it is an asset that provides a yield
of 5% per year. Figure1 shows currency futures quotes on January 8, 2007. The
quotes are USD per unit of the foreign currency.
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Figure 1: Foreign exchange futures quote on January 8, 2007.
We can synthetically create a forward contract by borrowing in one currency and
lending in the other. If we want to have 1 yen in the future, with the dollar price fixed
today we can pay today for the yen, and borrow in dollars to do so. The present value
of 1 yen in the future is e  rf T . Thus in order to have a 1 yen in the future one must pay
S0 e  rf T dollars, and we obtain this amount by borrowing. The required dollar
repayment is:
S0 e(r  rf )T
which is the same cash flow as a forward contract. If we offset this borrowing and
lending position with an actual forward contract, the resulting transaction is called
covered interest arbitrage.
To summarize, a forward exchange rate reflects the difference in interest rates
denominated in different currencies. Imagine that you want to invest $1 for a year.
You can do so by buying a dollar-denominated bond, or you can exchange the dollar
into another currency and buy a bond denominated in that other currency. You can
then use currency forwards to guarantee the exchange rate at which you will convert
the foreign currency back into dollars. The principle behind the pricing of currency
forwards is that a position in foreign risk-free bonds, with the currency risk hedged,
pays the same return as domestic bonds.
2.9. Futures prices and expected future spot prices
We refer to the market’s average opinion about what the spot price of an asset will be
at a certain future time as the expected spot price of the asset at that time. The
relationship between futures prices and expected spot prices is based on the
relationship between risk and expected return.
When you buy a stock you invest money that has an opportunity cost (it could
otherwise have been invested in an interest-earning asset) and you are acquiring the
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Leonidas Rompolis
risk of the stock. On average you expect to earn interest as compensation for the time
value of money. You also expect an additional return as compensation for the risk of
the stock – this is the risk premium. Algebraically, the expected return of the stock is:

r


r

compensation for time
compensation for risk
When you enter into a forward contract, there is no investment; hence, you are not
compensated for the time value of money. However, the forward contract retains the
risk of the stock, so you must be compensated for risk. This means that the forward
contract must earn the risk premium. If the risk premium is positive, then you must
expect a positive return from the forward contract. The only way this can happen is if
the forward price predicts to low a stock price. In other words the forward contract is
biased predictor of the future stock price.
We can see this algebraically. Let α be the expected return on a nondividend-paying
stock and let r be the annual interest rate. Consider a 1-year forward contract. The
forward price is:
F0  S0 (1  r)
The expected future spot price is:
E 0 (S1 )  S0 (1  )
Thus the difference between the forward price and the expected spot price is:
E 0 (S1 )  F0  S0 (1  )  S0 (1  r)  S0 (  r)
The expression α – r is the risk premium of the asset. The equation verifies that the
forward price is biased by the amount of the risk premium on the underlying asset.
For example, suppose that a stock index has an expected return of 15%, while the
risk-free rate is 5%. If the current index price is 100, then on average we expect that
the index will be 115 in 1 year. The forward price for delivery in 1 year will be only
105, however. This means that a holder of the forward contract will earn positive
profits, albeit at the cost of bearing the risk of the index.
This bias does not imply that a forward contract is a good investment. Rather, it tells
us that the risk premium on an asset can be created at a zero cost and hence has a zero
value. This result comes from the fact that if we buy any asset and borrow the full
amount of its cost – a transaction that requires no investment – then we earn the risk
premium on the asset. Since a forward contract has the risk of a fully leveraged
investment in the asset, it earns the risk premium (see also section 2.5).
Appendix
Suppose that the annualized daily compounded dividend yield is q. Suppose that we
have 1 share at time 0. Over the day 0 and day 1 we obtain a total dollar amount of
q
q
q
D1 
S0 . With this dollar amount we can buy
S0 S0 
shares of the
365
365
365
stock assuming that we are somewhere between 0 and 1 and thus we can
approximately use the stock price at 0 to buy the new shares.
q
Thus at time 1 we have 1 
shares of the stock. Over the day 1 and day 2 we
365
q
S1 per share. The total dollar amount is
obtain a dividend D 2 
365
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Leonidas Rompolis
q 
q  q


S1 . With this
1 
 D 2  1 

 365 
 365  365
q  q
q  q


S1 S1  1 
shares.
1 


 365  365
 365  365
dollar
amount
we
can
buy
2
q  
q  q
q 


 1 
Thus at time 2 we have 1 
  1 

 .
 365   365  365  365 
Continuing in the same manner we can demonstrate that at the end of the 1st year we
q 

will have 1 

 365 
365
q 

shares and in the end of T years we will have 1 

 365 
365T
.
Exercises
1. A 9-month long forward contract on a dividend paying stock is entered into when
the stock price is $100. The firm has announced that it will pay a dividend of $1.5
per share in 1 month and $2 in 6 months. The 1, 6 and 9-month risk-free rates are
2% and 3% and 3.5%, respectively.
(a) What are the forward price and the initial value of the forward contract?
(b) Three months later, the price of the stock is $90 and the 3 and 6-month riskfree rates are 2.5% and 3%, respectively. What are the forward price and the
value of the forward contract?
2. The S&P 500 index spot price is 1,050, the risk-free rate is 4%, and the dividend
yield of the index is 1%.
(a) Suppose you observe a 12-month forward price of 1,090. What arbitrage
strategy would you undertake? Calculate the profits of this strategy.
(b) Suppose you observe a 12-month forward price of 1,075. What arbitrage
strategy would you undertake? Calculate the profits of this strategy.
3. The 6-month risk-free rates in US and Europe are 2% and 3%, respectively. The
current exchange rate is $1.38 per Euro.
(a) Suppose that the 6-month futures price is $1.38. What arbitrage strategy
would you undertake? Calculate the profits of this strategy.
(b) Suppose that the 6-month futures price is $1.37. What arbitrage strategy
would you undertake? Calculate the profits of this strategy.
4. Suppose that the current price of the S&P 500 index is 800, and that the dividend
yield is 0. Assume that you can borrow money at 5.5% and that you can lend
money at 5%.
(a) Show that a cash-and-carry arbitrage is not profitable if the 1-year forward
price is less than 845.23, and that a reverse cash-and-carry is not profitable if the
forward price is greater than 841.02.
(b) Suppose that there is also a $1 transaction fee, paid at time 0, for going either
long or short the forward contract. Show that no-arbitrage exists in the market if
the 1-year forward price is between 846.29 and 839.97.
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Leonidas Rompolis
5. The file “FTSE 100 Data.xls” contains data for the FTSE 100 index futures
contracts for the period January 2003 to August 2003. Spreadsheet “Futures”
reports futures prices and respective days until maturity of the contract. It also
reports the LIBOR rates that correspond to the time-to-maturity of each contract.
Spreadsheet “Spot” reports the spot prices of the FTSE 100 index and daily
estimates of dividend yield.
(a) Filter the futures price data with respect to time-to-maturity, denoted as t.
Create 3 groups. The first consists of short-term futures with t  60 . The
second consists of medium-term futures with 60  t  180 . The third consists
of long-term futures with t  180 .
(b) For each one of these 3 groups calculate the no-arbitrage futures price using
the spot price, LIBOR rates and dividend yields data. Then calculate the
difference between the observed futures price and the no-arbitrage one.
(c) Calculate the average difference for these 3 groups over the sample period.
Explain your results.
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