2. Contents
Forward contracts..............................................................................................................................3
Definition..........................................................................................................................................3
ALTERNATIVE WAYS TO BUY A STOCK.................................................................................................3
PREPAID FORWARD CONTRACTS ON STOCK........................................................................................4
Pricing the Prepaid Forward by Analogy..........................................................................................4
Pricing the Prepaid Forward by Discounted Present Value...............................................................5
Pricing the Prepaid Forward by Arbitrage........................................................................................6
Pricing Prepaid Forwards with Dividends ........................................................................................7
1. Discrete Dividends...............................................................................................................7
2. Continuous Dividends..........................................................................................................8
FORWARD CONTRACTS ON STOCK..................................................................................................9
Does the Forward Price Predict the Future Spot Price?..................................................................11
Creating a Synthetic Forward Contract..........................................................................................12
Synthetic Forwards in Market-Making and Arbitrage.....................................................................14
3. Forward contracts
Definition:
Forward contract permit firms and investors to guarantee a price for a future purchase or
sale—are a basic financial risk management tool.
ALTERNATIVE WAYS TO BUY A STOCK
The purchase of a share of XYZ stock has three components:
(1) fixing the price,
(2) the buyer making payment to the seller, and
(3) the seller transferring share ownership to the buyer.
If we allow for the possibility that payment and physical receipt can occur at different times—
say,
time 0 and time T —then once the price is fixed there are four logically possible purchasing
arrangements: Payment can occur at time 0 or T , and physical receipt can occur at time 0 or T .
Let’s discuss these different arrangements.
1. Outright purchase. The typical way to think about buying stock. You simultaneously pay
the stock price in cash and receive ownership of the stock.
2. Fully leveraged purchase. A purchase in which you borrow the entire purchase price of
the security. Suppose you borrow the share price, S0, and agree to repay the borrowed
amount at time T . If the continuously compounded interest rate is r, at time T you
would owe erT per rupee borrowed, or S0e rT
3. Prepaid forward contract. An arrangement in which you pay for the stock today and
receive the stock at an agreed-upon future date. The difference between a prepaid
forward contract and an outright purchase is that with the former, you receive the stock
at time T. We will see that the price you pay is not necessarily the stock price.
4. 4. Forward contract. An arrangement in which you both pay for the stock and receive it at
time T , with the time T price specified at time 0.
Table below depicts these four possibilities, along with their customary names.
Four different ways to buy a share of stock that has price S0 at time 0. At time 0 you agree to a
price, which is paid either today or at time T. The shares are received either at 0 or T. The
interest rate is r.
From Table it is clear that you pay interest when you defer payment. The interesting question is
how deferring the physical receipt of the stock affects the price; this deferral occurs with both
the forward and prepaid forward contracts. What should you pay for the stock in those cases?
PREPAID FORWARD CONTRACTS ON STOCK
A prepaid forward contract entails paying today to receive something—stocks, a foreign
currency, bonds—in the future. The sale of a prepaid forward contract permits the owner to sell
an asset while retaining physical possession for a period of time.
The prepaid forward price is derived using three different methods: pricing by analogy, pricing
by present value, and pricing by arbitrage.
Pricing the Prepaid Forward by Analogy
Suppose you buy a prepaid forward contract on XYZ. By delaying physical possession of the
stock, you do not receive dividends and have no voting or control rights. (We ignore here the
value of voting and control.)
5. In the absence of dividends, whether you receive physical possession today or at time T is
irrelevant: In either case you own the stock, and at time T it will be exactly as if you had owned
the stock the whole time. Therefore, when there are no dividends, the price of the prepaid
forward contract is the stock price today. Denoting the prepaid forward price for an asset
bought at time 0 and delivered at time T as F0,T
P the prepaid forward price for delivery at time T
is
…………………….(5.1)
Pricing the Prepaid Forward by Discounted Present Value
We can also derive the price of the prepaid forward using present value: We calculate the
expected value of the stock at time T and then discount that value at an appropriate rate of
return. The stock price at time T, ST, is uncertain. Thus in computing the present value of the
stock price, we need to use an appropriate risk-adjusted rate.
If the expected stock price at time T based on information we have at time 0 is E0 (ST), then the
prepaid forward price is given by,
…………………….(5.2)
where α, the expected return on the stock, is determined using the CAPM or some other model
of expected returns.
How do we compute the expected stock price? By definition of the expected return, we expect
that in T years the stock will be worth
Thus, equation (5.2) gives,
For a non-dividend-paying stock, the prepaid forward price is the stock price.
6. Pricing the Prepaid Forward by Arbitrage
Classicalarbitrage describes a situation in which we can generate a positive cash flow either
today or in the future by simultaneously buying and selling related assets, with no net
investment of funds and with no risk. Arbitrage, in otherwords, is free money. An extremely
important pricing principle, which we will use often, is that the price of a derivative should
be such that no arbitrage is possible.
Here is an example of arbitrage. Suppose that the prepaid forward price exceeds the stock
price—i.e., FP
0,T > S0. The arbitrageur will buy low and sell high by buying the stock for S0 and
selling the prepaid forward for FP
0,T. This transaction makes money and it is also risk-free:
Selling the prepaid forward requires that we deliver the stock at time T, and buying the stock
today ensures that we have the stock to deliver. Thus, we earn FP
0,T − S0 today and at expiration
we supply the stock to the buyer of the prepaid forward. We have earned positive profits today
and offset all future risk.
Table 5.2 summarizes this situation.
Cash flows and transactions to undertake arbitrage when the prepaid forward price, FP
0,T
exceeds the stock price, S0.
Now suppose on the other hand that FP
0,T < S0. Then we can engage in arbitrage by buying the
prepaid forward and shorting the stock, earning S0 − FP
0,T . One year from now we acquire the
stock via the prepaid forward and we use that stock to close the short position. The cash flows
in the above table are simply reversed.
7. Throughout we will assume that prices are at levels that preclude arbitrage.
The transactions in Table 5.2 are the same as those of a market-maker who is hedging a
position. A market-maker would sell a prepaid forward if a customer wished to buy it. The
market-maker then has an obligation to deliverthe stock at a fixed price and, in order to
offset this risk, can buy the stock. The market-maker thus engages in the same transactions as
an arbitrageur, except the purpose is risk management, not arbitrage. Thus, the transaction
described in Table 5.2—selling the prepaid forward and buying the stock—also describes the
actions of a market-maker.
The no-arbitrage arguments we will make thus serve two functions: They tell us how to take
advantage of mispricing’s, and they describe the behavior of market-makers managing risk.
Pricing Prepaid Forwards with Dividends
When a stock pays a dividend, the prepaid forward price is less than the stock price. The owner
of stock receives dividends, but the owner of a prepaid forward contract does not. It is
necessary to adjust the prepaid forward price to reflect dividends that are received by the
shareholder but not by the holder of the prepaid forward contract.
1. Discrete Dividends.
To understand the effect of dividends, consider Stock A, which pays no dividend, and otherwise
identical stock B, which pays a $5 dividend 364 days from today. We know that the 1-year
prepaid forward price for stock A is the current stock price. What is the one-year prepaid
forward price for stock B?
The $5 dividend is paid just before the prepaid forward expiration date 1 year from today. Thus,
on the delivery date stock B will be priced $5 less than stock A. The prepaid forward price for
stock B should therefore be lower than that for stock A by the present value of $5.
In general, the price for a prepaid forward contract will be the stock price less the present value
of dividends to be paid over the life of the contract. Suppose there are multiple dividend
payments made throughout the life of the forward contract: A stock is expected to make
dividend payments of Dti at times ti ,i = 1, . . . , n. A prepaid forward contract will entitle you to
receive the stock at time T but without receiving the interim dividends. Thus, the prepaid
forward price is,
8. ………………..(5.3)
where PV0,ti denotes the time 0 present value of a time ti payment.
Example 1 Suppose XYZ stock costs $100 today and is expected to pay a $1.25 quarterly
dividend, with the first coming 3 months from today and the last just prior to the delivery of the
stock. Suppose the annual continuously compounded risk-free rate is 10%. The quarterly
continuously compounded rate is therefore 2.5%. A 1-year prepaid forward contract for the
stock would cost
Solution
The calculation in this example implicitly assumes that the dividends are certain. Over a short
horizon this might be reasonable. Over a long horizon we would expect dividend risk to be
greater, and we would need to adjust the discount rate in computing the present value of
expected dividends.
2. Continuous Dividends.
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. The
dividend yield is not likely to be fixed in the short run: When stock prices rise, the dividend yield
falls, at least temporarily. Nevertheless, we will assume a constant proportional dividend yield
in this case.
To model a continuous dividend, suppose that the index price is S0 and the annualized
daily compounded dividend yield is δ. Then the dollar dividend over 1 day is
Now suppose that we reinvest dividends in the index. Because of reinvestment, after T years
we will have more shares than we started with. Using continuous compounding to approximate
daily compounding, we get
9. At the end of T years we have approximately eδTtimes the shares we had initially.
Now suppose we wish to invest today in order to have one share at time T. We can
buy e−δT shares today. Because of dividend reinvestment, at time T , we will have eδT
more shares than we started with, so we end up with exactly one share. Adjusting the
initial quantity in this way in order to offset the effect of income from the asset is called
tailing the position. Tailing enables us to offset the effect of continuous dividends. We
will encounter the concept of tailing frequently.
Since an investment of e−δT S0 gives us one share at time T, this is the time 0 prepaid
Forward price for delivery at time T :
where δ is the dividend yield and T the time to maturity of the prepaid forward contract.
Example 2: Suppose that the index is $125 and the annualized daily compounded
dividend yield is 3%. The daily dollar dividend is,
Dividend = (0.03÷ 365) × $125= $0.01027
or a little more than one penny per unit of the index. If we start by holding one unit of
the index, at the end of 1 year we will have,
e0.03 = 1.030455
shares. Thus, if we wish to end the year holding one share, we must invest in
e−0.03 = 0.970446
shares. The prepaid forward price is,
$125e−0.03 = $121.306
FORWARD CONTRACTS ON STOCK
If we know the prepaid forward price, we can compute the forward price. The difference
between a prepaid forward contract and a forward contract is the timing of the payment for
the stock, which is immediate with a prepaid forward but deferred with a forward. Because
there is deferral of payment, the forward contract is initially costless for both the buyer and
seller; the premium is zero. Also, because payment is deferred, the forward price is just the
future value of the prepaid forward price:
10. ………………………5.5
This formula holds for any kind of dividend payment.
An important special case that we will use is that of a continuous dividend. In that case,
equation (5.5) becomes
Or
………………….5.6
The r in equation (5.6) is the yield to maturity for a default-free zero coupon bond with the
same time to maturity as the forward contract. For each maturity, there is an interest rate, r,
such that e−r(T−t) = P(t , T ), where P(t , T ) is the time t price of a zero-coupon bond maturing at
time T . We can therefore also write equation (5.6) as the price of the prepaid forward contract
divided by the price of a zero-coupon bond:
For forward contracts with different maturities, different values of r (and for that matter, δ) will
generally be appropriate. For simplicity we will write formulas as if there is a single interest rate
for all maturities, but in practice this is unlikely to be true. As is apparent from equation (5.6),
the forward price is generally different from the spot price. The forward premium is the ratio of
the forward price to the spot price, defined as,
………….……5.6
We can annualize the forward premium and express it as a percentage, in which case we have,
For the case of continuous dividends, equation (5.6), the annualized forward premium is simply
the difference between the risk-free rate and the dividend yield, r − δ. Occasionally it is possible
to observe the forward price but not the price of the underlying stock or index.
For example, the futures contract for the BSE index trades at times when the BSE is not open,
so it is possible to observe the futures price but not the stock price. The asset price implied by
the forward pricing formulas above is said to define fair value for the underlying stock or index.
Equation (5.6) is used in this case to infer the value of the index.
11. Does the Forward Price Predict the Future Spot Price?
It is common to think that the forward price predicts the future spot price. Equation (5.9) tells
us that the forward price equals the expected future spot price, but with a discount for the risk
premium of the asset. Thus, the forward price systematically errs in predicting the future stock
price. If the asset has a positive risk premium, the future spot price will on average be greater
than the forward price.
The reason is straightforward. 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 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 on a stock is,
……………………………….5.8
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 on average you must expect a
positive return from the forward contract. The only way this can happen is if the forward price
predicts too low a stock price. In otherwords the forward contract is a biased predictor of the
future stock price. This explains the discount by the risk premium in equation (5.9).
As an 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 on average earn positive profits, albeit at
the cost of bearing the risk of the index.
We can see the bias in another way by using equation (5.2). The prepaid forward price is the
present value of the expected future spot price. The forward price is the future value of the
prepaid forward price. Thus, we have
………………………….5.9
The forward price is the expected future spot price, discounted at the risk premium.
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 zero cost and hence has a zero value. Though this
seems surprising, it is a result from elementary finance 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
12. premium on the asset. Since a forward contract has the risk of a fully leveraged investment in
the asset, it earns the risk premium. This proposition is true in general, not just for the example
of a forward on a non-dividend-paying stock.
Creating a Synthetic Forward Contract
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.
In this discussion we will assume that dividends are continuous and paid at the rate δ,
and hence that equation (5.6) is the appropriate forward price. We can then create a synthetic
long forward 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 position on the
index is,
Payoff at expiration = ST − F0,T
In order to obtain this same payoff, we buy a tailed position in the stock, investing S0e−δT. 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 S0e(r−δ)T and we sell the stock for ST.
Table 3 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 5.4 demonstrates that we can go long a
forward contract and lend the present value of the forward price to synthetically create the
stock. The expiration payoff in this table assumes that equation (5.6) holds. Table 5
demonstrates that if we 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—the construction of which is
summarized in Table 5—is called the implied repo rate.
To summarize, we have shown that
Forward = Stock − zero-coupon bond…………………….5.10
Table 3
Demonstration that borrowing S0e−δT to buy e−δT shares of the index replicates the payoff to a
forward contract, ST − F0,T.
Table 4
13. Demonstration that going long a forward contract at the price F0,T = S0e(r−δ)T and lending the
present value of the forward price creates a synthetic share of the index at time T .
TABLE 5
Demonstration that buying e−δT shares of the index and shorting a forward creates a synthetic
bond.
We can rearrange this equation to derive other synthetic equivalents.
Stock = Forward + zero-coupon bond
Zero-coupon bond = Stock − forward
All of these synthetic positions can be reversed to create synthetic short positions.
TABLE 6
Transactions and cash flows for a cash-and-carry: A market-maker is short a forward contract
and long a synthetic forward contract.
14. TABLE 7
Transactions and cash flows for a reverse cash-and-carry: A market-maker is long a forward
contract and short a synthetic forward contract.
Synthetic Forwards in Market-Making and Arbitrage
Now we will see how market-makers and arbitrageurs use these strategies. Suppose a customer
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. Specifically, consider the transactions and cash flows in Table 6. The market-maker is
short a forward contract and long a synthetic forward contract, constructed as in Table 3. There
is no risk because the total cash flow at time T is F0,T − S0e(r−δ)T. All of the components of this
cash flow—the forward price, the stock price, the interest rate, and the dividend yield—are
known at time 0. The result is a risk-free position.
Similarly, suppose the market-maker wishes to hedge a long forward position. Then it is
possible to reverse the positions in Table 6. The result is in Table 7.
A transaction in which you buy the underlying asset and short the offsetting forward
contract is called a cash-and-carry. A cash-and-carry has no risk: You have an obligation to
deliver the asset but also own the asset. The market-maker offsets the short forward position
with a cash-and-carry. An arbitrage that involves buying the underlying asset and selling it
forward is called a cash-and-carry arbitrage. As you might guess, a reverse cash-and-carry
entails short-selling the index and entering into a long forward position.
If the forward contract is priced according to equation (5.6), then profits on a cash-and-
carry are zero. We motivated the cash-and-carry in Table 5.6 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., if F0,T > S0e(r−δ)T —then an arbitrageur or market-
maker can use the strategy in Table 6 to make a risk-free profit.
An arbitrageur would make the transactions in Table 7 if the forward were underpriced
relative to the stock—i.e., if S0e(r−δ)T >F0,T .
As a final point, you may be wondering about the role of borrowing and lending in
Tables 6 and 7. When you explicitly account for borrowing, you account for the opportunity
cost of investing funds. For example, if we omitted borrowing from Table 6, we would invest
15. S0e−δT today and receive F0,T at time T . In order to know if there is an arbitrage opportunity, we
would need to perform a present value calculation to compare the time 0 cash flow with the
time T cash flow. By explicitly including borrowing in the calculations, this time-value-of-money
comparison is automatic.
Similarly, by comparing the implied repo rate with our borrowing rate, we have a simple
measure of whether there is an arbitrage opportunity. For example, if we could borrow at 7%,
then there is an arbitrage opportunity if the implied repo rate exceeds 7%. On the other hand, if
our borrowing rate exceeds the implied repo rate, there is no arbitrage opportunity.
