Forwards, futures and swaps

Forwards and forward prices
In a forward contract, one party agrees to deliver a specified amount of a specified
commodity – the underlying asset – to the other at a specified date in the future (the
maturity date of the contract) at a specified price (the forward price). The commodity
may be a commodity in the narrow sense, e.g. gold or wheat, or a financial asset, e.g.
foreign exchange or shares. The price of the underlying asset for immediate (rather
than future) delivery is called the cash or spot price.
The party obliged to deliver the commodity is said to have a short position and
the party obliged to take delivery of the commodity and pay the forward price for it
is said to have a long position.
A party with no obligation offsetting the forward contract is said to have an open
position. A party with an open position is sometimes called a speculator. A party
with an obligation offsetting the forward contract is said to have a covered position.
A party with a closed position is sometimes called a hedger.
The market sets forward prices so there are no cash flows – no money changes
hands – until maturity. The payoff at maturity is the difference between forward
price, which is set contractually in the market at initiation, and the future cash
price, which is learned at maturity. Thus the long position gets STñFt,T and the short
gets Ft,TñST, where Tñt is the maturity, in years, of the forward contract (for
example, Tó1/12 for a one-month forward), ST is the price of the underlying asset
on the maturity date, and Ft,T is the forward price agreed at time t for delivery at time
T. Figure 1.4 illustrates with a dollar forward against sterling, initiated at a forward
outright rate (see below) of USD1.60. Note that the payoff is linearly related to the
terminal value ST of the underlying exchange rate, that is, it is a constant multiple,
in this case unity, of ST.
1.45 1.50 1.55 1.60
ST
0.05
0.10
0.05
0.10
payoff
forward rate
Figure 1.4 Payoff on a long forward.
No-arbitrage conditions for forward prices
One condition for markets to be termed efficient is the absence of arbitrage. The
term ‘arbitrage’ has been used in two very different senses which it is important to
distinguish:
Ω To carry out arbitrage in the first sense, one would simultaneously execute a set
of transactions which have zero net cash flow now, but have a non-zero probability
of a positive payoff without risk, i.e. with a zero probability of a negative payoff in
the future.
Ω Arbitrage in the second sense is related to a model of how asset prices behave. To
perform arbitrage in this sense, one carries out a set of transactions with a zero
net cash flow now and a positive expected value at some date in the future.
Derivative assets, e.g. forwards, can often be constructed from combinations of
underlying assets. Such constructed assets are called synthetic assets.
Covered parity or cost-of-carry relations are relations are between the prices of
forward and underlying assets. These relations are enforced by arbitrage and tell us
how to determine arbitrage-based forward asset prices.
Throughout this discussion, we will assume that there are no transactions costs
or taxes, that markets are in session around the clock, that nominal interest rates
are positive, and that unlimited short sales are possible. These assumptions are
fairly innocuous: in the international financial markets, transactions costs typically
are quite low for most standard financial instruments, and most of the instruments
discussed here are not taxed, since they are conducted in the Euromarkets or on
organized exchanges.
Cost-of-carry with no dividends
The mechanics of covered parity are somewhat different in different markets,
depending on what instruments are most actively traded. The simplest case is that
of a fictitious commodity which has no convenience value, no storage and insurance
cost, and pays out no interest, dividends, or other cash flows. The only cost of holding
the commodity is then the opportunity cost of funding the position.
Imagine creating a long forward payoff synthetically. It might be needed by a dealer
hedging a short forward position:
Ω Buy the commodity with borrowed funds, paying St for one unit of the commodity
borrowed at rt,T, the Tñt-year annually compounded spot interest rate at time t.
Like a forward, this set of transactions has a net cash flow of zero.
Ω At time T, repay the loan and sell the commodity. The net cash flow is
STñ[1òrt,T(Tñt)]St .
This strategy is called a synthetic long forward.
Similarly, in a synthetic short forward, you borrow the commodity and sell it,
lending the funds at rate rt,T,: the net cash flow now is zero. At time T, buy the
commodity at price ST and return it: the net cash flow is [1òrt,T(Tñt)]StñST.
The payoff on this synthetic long or short forward must equal that of a forward
contract: STñ[1òrt,T (Tñt)]STóSTñFt,T. If it were greater (smaller), one could make
a riskless profit by taking a short (long) forward position and creating a synthetic
long (short) forward. This implies that the forward price is equal to the future value
of the current spot price, i.e. the long must commit to paying the financing cost of
the position: Ft,Tó[1òrt,T(Tñt)]St .
Two things are noteworthy about this cost-of-carry formula. First, the unknown
future commodity price is irrelevant to the determination of the forward price and
has dropped out. Second, the forward price must be higher than the spot price, since
the interest rate rt,T is positive.
Short positions can be readily taken in most financial asset markets. However, in
some commodity markets, short positions cannot be taken and thus synthetic short
forwards cannot be constructed in sufficient volume to eliminate arbitrage entirely.
Even, in that case, arbitrage is only possible in one direction, and the no-arbitrage
condition becomes an inequality: Ft,TO[1òrt,T(Tñt)]St .