- Forward Prices
- Atomic Forward Prices
- Properties of Atomic Forward Prices
- Valuation Using Atomic Forward Prices
- Prices and Probabilities
- Return Swaps

We have used the term *price* to refer to an amount to
be paid *at the present time *for a time-state claim or
bundle of such claims. Not surprisingly, the magnitude of such a
value is specified at the present time. Thus party A might agree
to deliver a dollar next year if the weather is good, in return
for which party B delivers 0.285 dollars immediately.

It is, of course, possible to agree today to an exchange in which both receipts and payments will occur in the future. Forward interest rates represent the terms of such an arrangement. Similar procedures can be followed when contingencies are involved. For example, party A could agree to deliver 7 dollars next year if the weather is good while party B agrees to deliver 3 dollars next year if the weather is bad. In this case, both "sides" of the transaction are contingent and will take place (if at all) in the future.

Of particular interest are cases in which all payments are in
the future, but one involves payments that are not
state-dependent. For example, party A might agree to deliver one
dollar next year if the weather is good while party B agrees to
deliver 0.30 dollars next year *no matter what the weather has
been*. In this case we would say that the *forward price*
of a good weather dollar is 0.30 dollars delivered next year:
party B bought one good weather dollar for 0.30 dollars to be
delivered (with certainty) one year hence..

A forward price involves a future payment date. Thus in a multi-period setting, one could have a one-year forward price for a given set of time-state claims, a two-year forward price for the same set of claims, etc.. The first would indicate an amount that would have to be paid in one year to purchase the set of claims. The second would indicate an amount that would have to be paid in two years to purchase the set of claims, etc.. In each case, the price would be determined at the present time and the agreed-upon amount would have to be paid at the specified future time, regardless of the nature of ensuing events.

We have said that an *atomic price* is the present
value of a "pure security" -- i.e. a claim that pays
one unit of a stated commodity or currency at a specified time
and state of the world. For present purposes we focus on claims
that pay in currency terms. In particular, we assume that one
unit of such a claim pays $1 at the stated time if and only if
the state of the world occurs.

We define an *atomic forward price* as an amount that
must be paid with certainty for such a claim, with payment to be
made at the same time as the possible payment in question. Thus
in our example the atomic forward price for one good-weather
dollar is $0.30.

By extension, we may say that the forward price for a bundle of claims that share the same payment date, but differ only in the states of the world in which they are to be paid is the amount to be paid with certainty at the common date for which the bundle can be obtained.

Note that at time 0 there will be a forward price for, say, a
claim or combination of claims that will be paid (if at all) at
time 3. At time 1 there will be a potentially different forward
price for the same set of claims. However, contracts struck at
time 0 will require payment of the initial amount at time 3, even
though contracts newly negotiated at time 1 will require a
different amount. This means that a deal negotiated at time 0
that had a net present value of zero at the time may well have a
negative or positive net present value at time 1, depending on
the events that transpired in the first period. Realistic
accounting calls for both parties to adjust their books to
reflect the new value, thereby *marking to market* the
positions involved.

Arbitrage ensures that there is a very close relationship between prices and forward prices. Consider an economy in which a security promising $1 in year 1 if the weather is good commands a price of $0.285. Assume that an investment firm offers you such a security in return for a promise to pay $0.305 at the end of the year. Is this a good deal?

To obtain the answer one must consider the rate at which present (certain) dollars can be exchanged for certain future dollars. Assume that in this economy the one-period discount factor is 0.95. Thus one can exchange one certain future dollar for 0.95 present dollars. Alternatively, one can exchange 1/0.95, or 1.052632 certain future dollars for one present dollar (the one-period interest rate is 5.2632 percent per period). Assume that you wish to buy a good weather dollar security but pay for it at the end of the year. You could agree to pay $0.305 at the time to the investment firm. Alternatively, you could borrow $0.285 in order to buy such a security on the open market. You would, of course, have to repay the loan, which would require the payment of $0.285/0.95, or $0.285*1.052632 at the end of the year. But this is $0.300! Thus the investment firm was trying to make you agree to pay $0.305 in a year for something "worth" (obtainable elsewhere for) a promised payment of $0.300. In a market populated by astute Analysts, the investment firm would find that it had no takers for this product. If it were willing to take the other side of the offer, clever analysts could make money with neither risk nor investment via arbitrage between its terms and those available directly or indirectly in public markets.

Sooner or later, *arbitrage* will force equality
between the present price of a set of time-state claims and the
discounted forward price, using the appropriate discount factor
(or, equivalently default-free interest rate). Thus, if *fc(t)*
is the forward price to be paid at time t for the claim, *df(t)*
is the discount factor for time t, and *pc* is the present
price of the claim:

pc = df(t) * fc(t)

Given a present price, one can determine the appropriate forward price, or vice-versa, using the appropriate discount factor.

It is important to understand the precise meaning of an atomic
forward price. Consider the forward price of 0.300 for a dollar
in year 1 if the weather has been good. What *net*
payments would the forward purchaser of such an atomic claim have
to make? The answer depends on the weather.

If the weather is good:

Pay: $ 0.300 Receive: $ 1.000 ----- net $ 0.700

If the weather is bad:

Pay: $ 0.300 Receive: $ 0.000 ----- net - $ 0.300

In fact, the two parties could as well agree that if the weather is good, A will pay B $0.70, but if the weather is bad, A will receive $0.30 from B.

Forward atomic prices can, of course, be computed directly from (present) atomic prices and the discount factor. Assume that the present value of $1 if the weather is bad is $0.665. Then the atomic forward prices are:

good weather dollars: 0.285/0.95 = 0.300 bad weather dollars: 0.665/0.95 = 0.700

Note that they sum to 1.000. This is hardly a surprise. Consider the effect of buying one unit of every time-state claim. Such a bundle of claims will guarantee the receipt of $1.00 no matter what state may occur. It will also require the payment of an amount equal to the sum of all the corresponding forward atomic prices. If this sum is less than 1.000, one can get something for nothing by buying a package of equal amounts of all such claims. If it is more than 1.000, one can get something for nothing by selling such a package. Arbitrage will thus ensure that the sum will be 1.000. Thus:

The sum of the forward atomic prices for a given date must be 1.000.

The relationship between prices and forward prices allows one to value a set of time-state claims in two steps. First, all claims for a given time period are analyzed and their collective forward value determined. This is repeated for each time period. Finally, the resultant values are discounted, using the discount function, to obtain the overall present value.

Consider our previous example with two states of the world at
time 1 (**g** and **b**) and four
states at time 2 (**gg**, **gb **, **bg**,
and **bb**). The prices were given by **p**:

g b gg gb bg bb 0.2857 0.6667 0.0816 0.1905 0.1905 0.4444

The associated discount function **df** is thus:

Yr1 0.9524 Yr2 0.9070

The forward prices for year 1 are:

g b 0.30 0.70

and those for year 2 are:

gg gb bg bb 0.0900 0.2100 0.2100 0.4900

Now, consider the task of valuing the following set of claims **c**:

g 5 b 3 gg 15 gb 12 bg 11 bb 5

Given the atomic prices **p** the result can be
determined by simple matrix multiplication:

p*c = 11.2562

Here is the alternative.

First, compute the *forward value* of the time 1
claims:

State Payment Forward Forward Price Value g 5 .30 1.50 b 3 .70 2.10 ------- 3.60

Next, the forward value of the time 2 claims:

State Payment Forward Forward Price Value gg 15 0.09 1.35 gb 12 0.21 2.52 bg 11 0.21 2.31 bb 5 0.49 2.45 ------ 8.63

Finally, the discounted present value of both sets of claims:

Time Future Discount Present Value Factor Value 1 3.60 0.9523 3.4286 2 8.63 0.9070 7.8277 -------- 11.2562

Thus far, **nothing** has been said about *probabilities*.
This is just as well, for there is no "law of one
probability". Market participants can hold radically
different opinions concerning the probabilities of various states
of the world. No matter -- the markets will still function.
Prices will be set, valuation can be performed, replicating
strategies can be determined, packages of state-contingent claims
can be valued using atomic prices or the combination of atomic
forward prices and discount factors, and so on.

Despite these facts, there is a great temptation to interpret
atomic forward prices as probabilities. All such prices for a
given time sum to 1.0, as must any set of probabilities assigned
rationally to the states in question. The forward value of a set
of claims for a given time period could be interpreted as the *expected
value* of the payments if only the atomic forward prices were
probabilities. If so, one could argue that to value a set of
claims one only need discount the expected values, using riskless
rates of interest.

But there is no reason to expect that the atomic forward price
of a time-state claim equals the probability assigned to it by a
single market participant or even a consensus of market
participants. Quite the contrary. Hence it is dangerous to equate
an atomic forward price with any notion of the probability that
the associated state will occur. Nonetheless, many Analysts
accept the danger inherent in such a position, while recognizing
the fact that prices and probabilities need not be the same.
Commonly, they may use the term *risk-neutral probability*
instead of "atomic forward price", then argue that
valuation involves discounting (at riskless rates of interest)
the (pseudo-) expected payments at each period, with risk-neutral
probabilities used to calculate expected values.

The rationalization for this approach rests on two observations. As we will discuss subsequently, if investors were all risk-neutral and agreed on the probabilities of the various states of the world, the atomic forward price for a time-state claim would equal the agreed-upon probability of its occurrence. But if there is anything known about investors it is that they are risk-averse, not risk-neutral. Moreover, they do not all agree on probabilities. Hence atomic forward prices are not probabilities in any simple sense.

Despite these objections, those who use this nomenclature can still get the right answers. However, the economics of the situation are at best hidden from sight and may in many cases be overlooked entirely. Most importantly, it is easy to slip over the line and equate prices ("risk-neutral probabilities") with real probabilities. We attempt to avoid the confusion that such an approach can entail. Here, prices are prices and probabilities are probabilities. The relationships among them are complex and need to be addressed explicitly, which we do in other sections.

We conclude this section with yet one more example that illustrates that prices alone can provide all the needed answers for many important practical problems.

A very popular arrangement encouraged by financial engineers
can be termed a *return swap*. Consider the following
case. Investor A promises to pay investor B the *return*
on a *notional value* of $1 of a Stock, while B promises
to pay A the return on a value of $1 of a Bond. We utilize our
Stock that can increase by 26% or decrease by 4% in a year, and
the Bond that will increase by 5% for certain. In a one-period
setting, the return on an asset is simply the value-relative
minus 1. Thus the net cash flows are.

State Bond Stock A to B B to A good 0.05 0.26 0.21 -0.21 bad 0.05 -0.04 -0.09 0.09

The final columns of the table summarize the net payments between the counterparties to this swap in each of the possible states of the world. In practice only one payment is made: $0.21 from A to B if the weather is good or $0.09 from B to A if the weather is bad.

The obvious question: is this fair? The answer, shown below, is clearly yes. The present value of the amounts paid by A to B is precisely zero! Needless to say, so is the present value of the amounts paid by B to A.

Present State A to B price Value good 0.21 0.285 0.0598 bad -0.09 0.665 -0.0598 ------- 0.0000

This is a very general result, and depends in no way on the
simplified world analyzed here. As long as both parties will make
the required payments, *any* swap of the return on a
marketable security for the return on another one with equal
value will be "fair" (i.e. have zero net present value
at the time the deal is struck).

To see this, consider how one could "manufacture" the return on a security from other instruments. For simplicity, assume that the goal is to produce the dollar returns on the Stock in question: +$0.26 if the weather is good and -$0.04 if the weather is bad. The trick is to purchase one unit of the stock, and take out a loan that will require payment of $1.00 at the end of the year. The net results will then be:

Stock Loan State Value Payment Net good 1.26 -1.00 0.26 bad 0.96 -1.00 -0.04

as desired.

The cost of this strategy is $1.00 (for the stock) less the
cost of the loan, which is 1/1.05, or $0.9524. But the latter is
the discount factor for the time in question. Thus the present
value of a promise to receive the return on the stock at time
period 1 is **1-df(1)** which is, in turn, the *discount
on a 1-year zero-coupon bond*.

While we reached this conclusion for the Stock in our example, the result would have been the same if any other asset had been utilized, as a careful review of the argument will indicate. Moreover, with suitable modification, it would hold for other time periods. To generalize:

The present value of a guarantee to pay the return on an asset with a notional value of $X is the discount on a riskless loan that requires payment of $X at the end of the period over which the return is guaranteed.

Since a return swap involves exchange of two promises with the same present value, it is thus fair.

In practice, there is often some uncertainty concerning the
ability of one or both counterparties to make all promised
(contingent) payments. When this is the case, one or both present
values must be decreased (or promised payments increased) to
account for *credit risk*. Careful evaluation of such risk
is critical for success in the swap business.