11.2
A Note on Pricing Forward and Futures
Forward Contracts

ou learned about forward
contracts in Chapter 10, in topic 10.2, Interest
Rate Parity Relationship. In a
forward contract, two parties enter into an obligation to exchange one
asset for another at a price specified now but paid at a fixed time in the
future. No cash is exchanged at the
time of buying or selling a forward contract.
Therefore, to avoid arbitrage, the forward price must be set so that the
net present value of the contract is zero.
You can see how the
arbitragefree forward price is determined if you consider the following two
alternatives. In alternative A
you buy a forward contract at time 0, and at T
in the future you receive the asset and pay $F
cash. In alternative B you purchase the asset today for $S (its price today) by borrowing $S cash for time period T
at the riskfree interest rate r.
At time T you repay the loan.
In either case, you do
not need any of your own money today. At
time T, your position under
alternative A is:
$F
cash, +1 unit of the asset
while under
alternative B, it is:
In either case, you have
the asset at time T.
Therefore, the two alternative ways of buying the asset must cost the
same. Otherwise, you can make
arbitrage profits. This means:
A variation on this is
that it costs you something to hold the asset until period T. This cost is called
a “carrying cost.” For example,
if the asset is a commodity, such as wheat, you will generally have to pay to
store the commodity. Such costs are
easily accommodated. Suppose you
pay a lump sum at time T; then, we
must have
If you have to pay the
cost “continuously,” say, at rate b,
then we must have
To see why this
relationship must hold, suppose that the forward price F is strictly greater than SerT (a
similar argument applies if it is strictly less).
In this case, you would
want to sell the forward contract for $F,
borrow $S for time T,
and buy the asset at the current spot price.
By the nature of a forward contract, you will not receive any cash now
for this transaction, and we will assume that holding costs are zero.
At the end of the contract's life, however, you must deliver one unit of
S for the forward price $F.
Thus, at this time you deliver your one unit of S,
receive $F, and repay the loan.
This chain of
transactions leaves you with
cash in hand, which by
our original assumption is strictly positive.
In Chapter 10, topic
10.2, Interest Rate Parity Relationship,
the general form for the arbitragefree valuation of a currency forward contract
is derived as an extension of this model to accommodate the interest rates for
each currency:
Futures Contracts
The primary difference
between a futures contract and a
forward contract, from a valuation perspective, is that the futures contract is marked
to market on a daily basis. What
this means is that the futures price
is reset so that the value of the futures contract is zero.
All outstanding contracts are then adjusted by adding or subtracting cash
from the accounts of the contract holders.
For example, suppose you
sold one futures contract at the futures price F and subsequently the spot price, S, rises to S '.
Your futures contract is now worth a negative amount.
To see why, let F ' be the new
forward price (the future value of the spot price assuming carrying costs are
zero):
When your contract is
marked, your account has the difference
Similarly, if you make
money, your account has the difference added to it.
If interest rates are
stochastic, it is not so easy to value futures contracts (or options on
futures). This is because the
arbitrage argument outlined in the previous section no longer applies;
alternative A may cost you more or less than alternative B, depending on the
path of interest rates.
In practice, a commonly
made assumption is that for the purposes of valuing futures and options on
futures, interest rates can be taken to be deterministic.
It turns out that in this case, forward and futures prices must be the
same, and we can value options on forwards and futures in the same way.
The argument is due to Cox, Ingersoll, and Ross (1981), and we show you a
twoday version of their argument when the interest rate is constant.
Assume delivery takes
place on day 2. Let F0
be the current futures price, and let F1
and F2 be the futures prices at the end of days 1 and 2.
The futures strategy is to buy er
futures contracts at the end of day 0 and to buy the same number again at the
end of day 1. Consider what marking to market does.
.
The future value of
these additions/subtractions (at the end of day 2) is
compounded once
plus
which comes out to
But F2 is the
futures price at time of delivery, and must equal the spot price on day 2, S2,
so the net result is
Now, suppose you
buy e^{2r }forward contracts at the forward price, say G0. At time 2, you
would receive the asset, worth S2,
and would pay G0 for each
contract, yielding you a value of
If F0 > G0,
you can pursue the following arbitrage strategy: sell the first strategy (i.e.,
go short the sequence of futures trades) and buy the second strategy (i.e., go
long the forwards). You will make
money no matter what S2
turns out to be.
The argument can be
extended to cover the case where interest rates follow a deterministic path; you
simply adjust the number of contracts you buy to make the end result the same.
Like many others, we
assume that for the purposes of valuing futures contracts, interest rates can be
taken to be deterministic. This
allows us to value not only futures contracts but also options on futures (or
forwards). This model was first
developed by Black (1976). Online,
click on the Pricing of Options on Futures
to see Black's options on futures pricing model.