- Expanding the World
- Money
- Real and Nominal Interest Rates
- Multiple Commodities
- Multiple States
- Incomplete Markets
- Hedging at Minimum Cost
- Completing a Market
- Multiple Times
- Dynamic Strategies
- Model Risk
- Options
- Derivatives

The Apple economy has served us well, but it is time to consider more complex worlds.
We begin with situations in which there are other commodities and, most importantly, *money*.
We then consider circumstances in which more than two states of the world can occur in a
single period. Finally, we expand the analysis to cover situations in which there are more
than two time periods.

Standard definitions assign the term "money" to instruments that are *legal
tender* within a political jurisdiction. In principle, one must accept such
instruments in the settlement of transactions. Currency and deposits against which checks
can be written are usually considered money. Assets that can quickly, easily and cheaply
be turned into money are often termed "near-money". For our purposes, money can
be thought of as a *medium of exchange*.

We will generally use *dollars* as our standard monetary unit in examples. One
may think of these as U.S. Dollars, Australian Dollars, Hong Kong Dollars, or any other
such currency. For notation, we follow the standard practice of preceding an amount with
the identifying symbol. Thus $1.50 represents 1.50 dollars. For symmetry, we will do the
same for apples: hence, A2.50 represents 2.50 apples.

In most economies, conditions of trade are stated in monetary units. Moreover, most
trades involve a swap in which at least one side involves money *per se*. Thus we
trade money for apples (buy apples), trade money today for money a year from now (e.g. buy
a one-year Treasury bill), trade money today for apples next year, etc.. If one wishes to
trade today's apples for next year's oranges, it may be most efficient to sell the apples
today (trade today's apples for today's money), invest the proceeds (trade today's money
for next year's money), and use the proceeds to purchase oranges next year (trade next
year's money for next year's oranges).

The number of dollars for which a commodity can be traded at any given time is
generally termed its *price*. In some contexts it is desirable to use a more
precise term: the *spot price* of a commodity is an amount determined and paid
contemporaneously with the delivery of that commodity.

The spot price of a commodity will depend on both the amount of the commodity and the
amount of money available at the time. Other things equal, the greater the amount of money
relative to the amount of a commodity, the higher will be its price. *Inflation*
(rising prices) is often attributed to "too much money chasing too few goods".
Central banks attempt to control national money supplies to avoid excessive inflation (and
deflation). However, other objectives and outside influences may compromise such good
intentions. In practice, there is by no means a simple one-to-one relationship between the
money supply and prices. As economies become more intertwined it becomes more and more
difficult for a government or central bank to manage any element of a national economy,
including its prices. And, of course, even if the general level of prices in an economy
remains constant, there will be changes in relative prices, as dictated by changing demand
and supply conditions.

To illustrate the behavior of a monetary economy, we begin with a world with only apples and money. As before, there are two periods and two future states of the world (good and bad weather). Initially, we assume that the monetary authorities are able to adjust the money supply so that there is more money when there are more apples and less when there are fewer and that this adjustment will succeed in keeping the price of an apple constant at $0.50. As in the earlier examples, we assume that the price of 1 good weather apple is 0.285 present apples and that the price of 1 bad weather apple is 0.665 apples.

Consider first all the possible trades between the present and time 1 if the weather is good:

Time 0 Time 1 (good) A0.285 ---------------- A1 | | | A1 = $0.50 | A1 = $0.50 | | $0.1425 $0.50

If apples can be traded as shown in the top portion of the diagram, then it is possible to trade $0.1425 today for $0.50 next year if the weather is good. This follows from the fact that knowledge of the state of the world resolves all uncertainty at any specific time. Thus at time zero we know that if the weather turns out to be good, the price of an apple will be $0.50 at time 1. Hence $0.1425 can be converted to 0.285 apples today, those apples can be used to purchase a claim for 1 apple next year if the weather is good, and we know in advance that that apple can be converted to $0.50 if that state of the world obtains.

In practice, the economy is more likely to look like this:

Time 0 Time 1 (good) A0.285 A1 | | | A1 = $0.50 | A1 = $0.50 | | $0.1425 -------------- $0.50

Explicit markets will exist for buying and selling commodities at each time period,
with transactions involving *time* and/or *uncertainty* conducted in
monetary units. Thus if one wants to swap today's apples for next year's apples if the
weather is good, one might sell A0.285 apples, obtain $0.1425, use this amount to purchase
a claim for $0.50 if the weather is good, and plan to use that amount to purchase 1 apple
when and if the state of the world obtains.

Well and good, but what if the price of a commodity in a future time and state is
expected to differ from that of today? Assume that if the weather is good, the price of an
apple is expected to rise to $0.60 due to an increase in the money supply, the *velocity*
at which money circulates, or both. In this case we would have:

Time 0 Time 1 (good) A0.285 A1 | | | A1 = $0.50 | A1 = $0.60 | | $0.1425 -------------- $0.60

Note that the current apple price of one good weather apple remains at A0.285. However,
the current dollar price of one good weather dollar is $0.1425/$0.60, or $0.2375. We term
the former a *real* (apple) exchange rate and the latter a *nominal* one.
When future commodity prices differ from current prices, there will be disparities of this
sort. However, arbitrage will insure that there is a close relationship among commodity
prices, real exchange rates and nominal exchange rates.

To complete this latter example, assume that if the weather is bad the price of apples will remain at $0.50. Thus:

Time 0 Time 1 (bad) A0.665 A1 | | | A1 = $0.50 | A1 = $0.50 | | $0.3325 --------------- $0.50

The dollar price of one dollar if the weather is bad is $0.3325/$0.50, or $0.665; in this case the apple and dollar exchange rates are the same.

In the current example, there are two discount factors:

Real 1 good weather apple = 0.285 present apples 1 bad weather apple = 0.665 present apples ------------------- -------------------- 1 future apple = 0.950 present apples Nominal 1 good weather dollar = 0.2375 present dollars 1 bad weather dollar = 0.665 present dollars ------------------- -------------------- 1 future apple = 0.9025 present apples

Thus the *real discount factor* is 0.950, while the *nominal discount factor*
is 0.9025.

Closely related to the concept of a discount factor is that of the default-free *interest
rate*. For a case involving only the present and a future period, the rate may be
expressed on a per-period basis and calculated simply. If the discount factor is **d**,
then one unit will "grow to" 1/d in one period. Thus, given a real discount
factor of 0.95, one apple will grow to 1/0.95, or 1.0526 (approximately) apples in one
year. We say that the associated *interest rate* is 0.0526, or 5.26 percent per
year. Thus:

1+i = 1/d or: i = (1/d) - 1 equivalently: d = 1/(1+i)

In our example, the real interest rate is 5.26%, while the *nominal interest rate*
is approximately 10.80%: (1/0.9025)-1. In a potentially inflationary environment, nominal
interest rates will be higher than real interest rates, with the difference larger, the
greater the likely degree of inflation.

In the real world there are, of course, many different types of commodities (there are also different types of currencies, but we leave this complication for a later discussion). Formally, the time-state approach assumes that once the state of the world is known, all uncertainty is resolved concerning contemporaneous commodity prices. Thus markets need only (!) exist for each commodity and money at a given time and for claims for money across time and possible states of the world.

With multiple commodities, the simple notion of a real exchange rate breaks down, and
with it the notions of real discount factors and real interest rates. For example, the
real interest rate expressed in oranges may differ from that expressed in apples, kumquats
or whatever. Economists attempt to get around this problem by using the price of a
pre-defined basket of goods and services to compute a *price index*. They then
convert nominal exchange rates to real rates using the price of such a basket of goods.
Such undertakings are fraught with hazard. A basket is unlikely to be fully representative
of the purchasing habits of a given individual or institution. If the basket's composition
is held fixed, the change in cost will likely overstate the cost of obtaining a constant
degree of satisfaction, since adaptation to a new set of relative prices is not taken into
account. Finally, it is difficult to fully take into account changes in quality when
attempting to determine a change in the "cost of living" (or producing) at a
given level of happiness (or efficiency).

Despite these problems, price indices are important for financial analysis.
Accordingly, governmental agencies compute and publish various versions designed to
represent the costs faced by representative consumers and producers. Most countries have
established *consumer price indices* as well as *producer price indices*.
More general measures are those used for computing overall national statistics, in
particular *gross domestic product deflators*, employed to estimate changes in the
real levels of domestic production of economies.

Any price index can be used to estimate a real counterpart for a nominal value. In practice, Analysts usually employ a consumer price index (CPI) designed to represent (as best possible) the buying habits of a typical member of an economy.

Thus far we have assumed that from one period to the next there are only two possible states of the world (specifically, good weather and bad weather). For many applications this stretches credulity. Over a year there can be good weather, fair weather, bad weather, plagues, pestilence, and so on. If an entire economy is to be modeled, one may need to consider a multiplicity of possible outcomes.

Imagine a world in which there are two periods (now and a year from now) and three
possible states of the world (Good Weather, Fair Weather and Bad Weather). As before,
there is a *Bond* and a *Stock*. All values are stated in dollars. The
payment matrix is given by **Q**:

Bond Stock Good Weather 20 43 Fair Weather 20 35 Bad Weather 20 28

and the security prices by **ps**:

Bond Stock 19 30.875

What are the atomic prices?

Our general rule is, of course

p = ps*inv(Q)

But it is impossible to take the inverse of Q, since it is not square. The number of rows is equal to the number of possible states of the world. To be able to invert the matrix we need as many columns (securities) as there are rows (states of the world). In this case we need one more security.

Assume some research turns up a *convertible bond*. Such an instrument is a bond
with promised payments that can, on the holder's demand, be converted to a common stock
with equity interest. Since one should only undertake such a conversion when the equity is
worth more than the bond, we can write the cash flows associated with the various states
of the world assuming *optimal exercise* of the option to convert. Assume that
doing so gives the payment matrix **Q**: :

Bond Stock Convertible Good Weather 20 43 35 Fair Weather 20 35 25 Bad Weather 20 28 25

If the convertible is selling for $24, we have the security price vector **ps**:

Bond Stock Convertible 19 30.875 24

We can now compute the atomic prices **p = ps*inv(Q)**. To four decimal
places they are::

Good Fair Bad Weather Weather Weather 0.0250 0.5571 0.3679

The discount factor is, as always, **sum(p)**. In this case:

sum(p) = 0.95

which is not surprising, given the presence of the same riskless bond as used in the earlier examples.

With three states of the world, three securities are needed to *"span the
space"*. However, to do so, the securities must be sufficiently different.
Consider, for example, the following payment matrix **Q**:

Bond Stock Security X Good Weather 20 43 21.5 Fair Weather 20 35 17.5 Bad Weather 20 28 14.0

If you were to try to take the inverse of this matrix, the results would be (at the very least) a warning message. MATLAB would say something like:

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.648137e-019

The problem is not difficult to discern -- every payoff from Security X is precisely
half that from the Stock. It is thus not different enough to complement the other two
securities and allow construction of portfolios that can replicate any desired set of
payments across the three states. The securities are not "different enough" --
they do not *span the state space*.

To see what the latter expression means, consider again our earliest example of the Apple Tree Bond and Stock. The diagram showing the opportunity set for contingent payments per dollar invested is repeated below, with two arrows added.

The point shown for the Stock depends on three values -- its payments in each of the two states of the world and its current price. Given the payment vector, the price will determine the actual location of the point, but it will lie on the vector shown by the arrow through the current point, no matter what the price may be.

Similarly, the Bond will plot at some point along the vector shown by the arrow through its current point. As long as the arrows are distinct ("different"), the securities will plot at two different points and support an opportunity set with a boundary such as that shown by the line in the diagram.

Imagine the consequences if the securities fell on the same vector (arrow). If they were priced to plot at the same point, it would clearly be impossible to use them to obtain any other combination of payments. If they plotted at different points, one could take a short position in one and a long position in the other and make a potentially infinite amount of money with no risk and no investment! The latter case is of course implausible, and both fail our test..

In general, if there are **S** states of the world in a given time period,
**S** securities that plot on different "arrows" in the state space
are required. Since the locations of the arrows depend only on the payment vectors, it is
the set of such vectors (our matrix **Q**) that must meet this condition. The
test is simple: if **Q** can be inverted, the securities are sufficiently
different. If it cannot, they are not.

If an available set of securities spans a state space, we say that the markets are *complete*.

If states of the world are defined very narrowly, the number of possible states of the
world is very large. The number of securities (broadly construed) is also large, but
almost certainly smaller than the number of possible states. Thus markets are *incomplete*
in a global sense. On the other hand, for many applications it is sufficient to define
states broadly. For example, assume that an Analyst wishes to value and/or hedge an
investment product that has payments tied to the level of a stock index. For such an
analysis, there is no need to differentiate between the state "Stock Index level =
$500 and sailing conditions are good" and the state "Stock Index level = $500
and sailing conditions are bad". The broader state "Stock Index level =
$500" suffices, for it resolves all the uncertainty that is relevant for the issue at
hand. While securities may not span a detailed space, they may do so very well for a more
aggregated one.

Consider the following payment matrix **Q**:

Bond Stock Good Weather 20 43 Fair Weather 20 28 Bad Weather 20 28

and price vector **ps**:

Bond Stock 19 35

Each security provides the same payment in the state *Fair Weather* as in the
state *Bad Weather*. If one is interested only in payment patterns that also have
this characteristic, the problem may be reduced to one involving two states. Thus, we have
**Q**:

Bond Stock Good Weather 20 43 Fair or Bad Weather 20 28

and can compute the atomic prices **p = ps*inv(Q)**. They are::

Good Fair or Bad Weather Weather 0.56 0.39

The latter price is, in effect, the sum of two prices: the price of $1 if the weather
is fair and the price of $1 if the weather is bad. We are able to measure the sum but have
no way to know what the value of each of the components might be. The markets are *sufficiently
complete* to price or replicate any payment pattern in which the same amount is to be
paid in the two states (Fair and Bad Weather). However, if someone asks for a payment
pattern in which a different amount is to be received in Fair Weather than in Bad Weather,
it will be impossible to find a replicating strategy involving the Bond and the Stock in
question.

This example is of considerable practical importance. Payment patterns that can be
replicated with existing securities can be priced with considerable accuracy. Moreover, an
Investment Firm can offer products with such patterns by taking an offsetting position in
the appropriate replicating strategy. When this is done, the product is said to be *fully
hedged* and there is in principle no risk associated with it other than uncertainty
about the future state of the world.

More interesting but more problematic are investment products that offer payment patterns not available with combinations of existing securities. Such products cannot be fully hedged by the provider, nor can their prices be established definitively using the prices of other securities. Consider an Investor who asks an Investment Firm to create an investment product with the following payments:

Good Weather 40 Fair Weather 30 Bad Weather 20

What should the Investment Firm charge? And how might it hedge as much of the associated risk as possible?

The problem is, of course, the fact that no matter what combination of the Bond and the Stock is chosen, the net payments will be the same in Fair Weather and Bad Weather. To be absolutely certain that no net outlay might be required, the firm would have to select a combination of securities that would pay 40 in Good Weather and 30 in Fair or Bad Weather. In this case:

ps = Bond Stock 19 35 Q = Bond Stock Good Weather 20 43 Fair or Bad Weather 20 28 c = Good Weather 40 Fair or Bad Weather 30 n = inv(Q)*c: Bond 0.5667 Stock 0.6667 price = ps*n: 34.10

Thus it would cost $34.10 (price) to purchase a portfolio (n) that would cover all outflows. Note, however, that in the event that the weather is Bad, the portfolio will provide $30 while the firm would be obligated to pay out only $20, leaving $10 as profit. Thus the Investment Firm would be delighted to sell the product for a price of $34.10. Moreover, if it did so and undertook the hedge (n), the Investor would be assured of receiving the promised payments under all circumstances.

The approach used in the previous example may be generalized by formulating the problem in a manner that allows for the number of securities to be less than, equal to or greater than the number of states of the world. The goal is to minimize the cost of meeting or exceeding the required payment in each state of the world. Using our previous notation:

select: n to minimize: ps*n subject to: Q*n >= c

In this formulation, **n** {securities*1} is the vector of *decision
variables*, **ps** {1*securities} contains the coefficients of the linear
*objective function*, or *minimand*, **Q** {states*securities}
contains the coefficients of the left-hand side of the *constraint set*, and **c**
{states*1} contains the coefficients of the right-hand side of the constraint set.

The matrix representation of the constraint set is straightforward, The left-hand side **Q*n**
is a {states*1} vector, as is the right-hand side **c**. The matrix
inequality indicates that each element of the left-hand vector (amount available to be
paid) must be greater than or equal to the corresponding element of the right-hand vector
(amount required to be paid).

Since this problem involves a linear objective function and linear inequality
constraints, it is a member of the class of *linear programming problems* and can
be solved using any of a number of *algorithms* (procedures) designed for such
tasks. MATLAB's optimization toolbox includes a function named *lp* for this
purpose. The simplest use is described in MATLAB as follows:

x = lp(f,A,b) solves the linear programming problem:

min f'x subject to: Ax <= b x

This is almost precisely what we need. However, the inequality is reversed.
This is easily overcome, for:

Q*n >= c

is the same as:

-Q*n <= c (for example: 3>=2 and -3

<=-2)

Thus the problem can be solved with the following MATLAB expressions:

n = lp(ps,-Q,-c);

price = ps*n;

While our current interest is in the use of the linear programming formulation in an
incomplete market setting, it can also be used in a complete market. In such a case,
the procedure can produce a portfolio that will achieve the required set of payments
exactly. If more securities are included than there are states, there will typically be
multiple ways of meeting the goals; however, if the markets are arbitrage-free, all such
portfolios will have the same cost.

Linear programming algorithms can provide a set of *Lagranian multipliers*, each
of which indicates the change in the objective function (here, cost) per unit change in
one of the right-hand side coefficients. In this instance, each such multiplier for
constraints that are *binding* is the atomic price for a state -- how much it would
cost to have one more dollar paid in that state.

Here is the MATLAB description of the procedure used to obtain these multipliers:

[x,LAMBDA]=lp(f,A,b) returns the set of Lagrangian multipliers, LAMBDA, at the solution.

To obtain the set of atomic prices and the hedge portfolio, one could use the MATLAB
expression:

[n p] = lp(ps,-Q,-c);

We return now to the problem faced by the Investment Firm. A potential client wants a
product that will pay **c**:

Good Weather 40

Fair Weather 30

Bad Weather 20

But the only existing securities are a Bond and a Stock with payments **Q**:

Bond Stock Good Weather 20 43 Fair Weather 20 28 Bad Weather 20 28

and pricesps:

Bond Stock 19 35

The firm has run its linear programming algorithm and found that for $34.10 it can meet
its obligation in every state of the world, but with $10 left over if the weather is Bad.
Surely, it figures, this is worth something (but at most 0.39*10 = $ 3.90,
according to our previous calculations). Assume that after some thought, it offers
the product for $32.00. Moreover, it is willing to "make a market" and
"take either side", that is, buy or sell the product at that price.

We now have three securities, with payment matrix, **Q**:

Bond Stock Product Good Weather 20 43 40 Fair Weather 20 28 30 Bad Weather 20 28 20

and price vector **ps**:

Bond Stock Product 19 35 32

The market is now complete, with atomic prices **p = ps*inv(Q)**:

Good Fair Bad Weather Weather Weather 0.56 0.18 0.21

In effect, the Investment Firm has priced the Fair Weather and Bad Weather claims at
$0.18 and $0.21, respectively, resolving the indeterminacy of the split of the prior known
value of $0.39 between the two claims.

In the real world, as in this example, an Investment Firm, by offering a new product with sufficient guarantees of payment in all circumstances, can provide an important service by making the capital markets more complete. In this case the availability of such a product fully completed the market, since the Bond, the Stock and the new Product completely span the space of relevant states of the world. Any desired set of payments across the three states can be replicated with some combination of these three instruments.

Even if motivated only by greed and cupidity, Investment Firms can provide significant social services and move markets closer and closer to the idealized ones described in works such as this.

Practical applications of the time-state approach often dispense with the convenient
fiction that the world ends after one period. It is time for us to do so as well.
Following practice, we focus on cases in which a variable of interest can move in
one of two directions in each time period. To keep things as simple as possible, we
allow for two future time periods (times 1 and 2), in addition to the present (time 0).

The *tree* of possible states of the world now has seven *nodes*:

Instead of numbering the nodes sequentially we use letters for all but time zero. The
number of letters indicates the time period (thus state **gg** is at time
period 2, since there are two letters). The sequence of letters indicates the path taken
to reach the node (thus **gb** indicates a good branch followed by a bad
one).

A diagram such as this is sometimes termed a *lattice*. Since only two branches
emanate from each node, the underlying relationship is often termed a *binomial process*.

A security or Investment Product is represented with a set of values or cash flows in such a tree. We start with a two-period zero-coupon Bond that grows in value by 5% per period. Its initial value is $1.00. The values of a Bond at the nodes are shown below:

Our second security is a Stock that pays no dividends. Its price increases 26% in good times but falls to 96% of its prior value in bad times. Its initial value is also $1.00. The values at the nodes are shown below:

In this case **S**, the number of future states of the world equals six.
Our previous discussion indicated that six different securities would be required to span
this space and hence allow replication and valuation of Investment Products. This remains
true if the term "security" is expanded to include a *planned acquisition of
a security in the future*. Taking this view, six distinct elemental combinations of
payments can be provided using the two traded instruments. The associated purchases and
sales are as follows:

- B0: Buy a bond today; sell it at the end of period 1
- S0: Buy a stock today; sell it at the end of period 1
- Bg: At period 1, if the state is
**g**, buy a bond; sell it at the end of period 2 - Sg: At period 1, if the state is
**g**, buy a stock; sell it at the end of period 2 - Bb: At period 1, if the state is
**b**, buy a bond; sell it at the end of period 2 - Sb: At period 1, if the state is
**b**, buy a stock; sell it at the end of period 2

Only the first two strategies involve an outlay at the present. The latter involve
outlays (negative cash flows) at future times under some circumstances, but none today.
The associated payment matrix **Q** is:

B0 S0 Bg Sg Bb Sb g 1.05 1.26 -1.00 -1.00 0 0 b 1.05 0.96 0 0 -1.00 -1.00 gg 0 0 1.05 1.26 0 0 gb 0 0 1.05 0.96 0 0 bg 0 0 0 0 1.05 1.26 bb 0 0 0 0 1.05 0.96

The price vector **ps** is:

B0 S0 Bg Sg Bb Sb 1.00 1.00 0 0 0 0

To find the atomic prices, we proceed as always to find **p = ps*inv(Q)**:

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

To see how a multiple-time approach can be used in a practical situation, consider the
following Investment Product:

At time 2, Investment Firm will pay the holder an amount equal to:

$1.50 if the Stock is worth more than $1.50

$1.00 if the Stock is worth less than $1.00

The value of the Stock otherwise

What is this *collar* around the price of the Stock worth? Is there some
other way that an Investor could obtain the same results?

To answer these questions, construct a vector (**c**) with the cash flows
associated with the product:

g 0 b 0 gg 1.50 gb 1.2096 bg 1.2096 bb 1.00

Next, compute the value= p*c:

1.0277

and the replicating portfolion = inv(Q)*c:

B0 0.2853 S0 0.7423 Bg 0.2670 Sg 0.9680 Bb 0.3136 Sb 0.6987

While we may term this a *portfolio*, it would usually be considered a *dynamic
strategy*. It calls for an initial purchase of $0.2853 of Bonds and $0.7423 of
Stocks (for a total cost of $1.0277). If the weather turns out to have be good at
the end of the first period, the Bonds will have grown to 1.05*$0.2853, or $0.2996, while
the Stocks will have grown to 1.26*0.7423, or $0.9354, giving a total portfolio value of
$1.2350. The strategy calls for this portfolio to be sold and a portfolio with
$0.2670 of Bonds and $0.9680 of Stocks purchased. The cost will be precisely equal
to the proceeds obtained from the sale of the initial positions, as shown below:

Initial Revised Difference Bond 0.2996 0.2670 -0.0326 Stock 0.9354 0.9680 0.0326 ------- -------- -------- 1.2350 1.2350 0.0000

In fact, of course, it would only be necessary to sell $0.0326 of Bonds and purchase
$0.0326 of Stocks to implement the needed change.

If the weather turns out to have been bad, the stock position at the end of the year will only be worth 0.96*$0.7423, or $0.7127. The situation would then be the following:

Initial Revised Difference Bond 0.2996 0.3136 0.0140 Stock 0.7127 0.6987 -0.0140 ------- -------- -------- 1.0123 1.0123 0.0000

In this case, $0.0140 of Stocks would be sold and the proceeds used to purchase $0.0140
of Bonds.

In principle, any set of time and state-contingent cash flows can be replicated with
any set of securities that spans the relevant space of time-state claims. Moreover,
if there are only two possible branches at each node in the tree representing the
underlying process, planned acquisition and sale of two different securities at each
intermediate future time period will suffice to span the entire space.

What can go wrong? Two things. First, a counterparty can default, in whole or in part,
on an obligated payment. Second, the model may be wrong. The possibility of the latter is
known as *model risk*.

Two examples will illustrate the type of dangers lurking behind such arrangements.

Assume that the tree drawn in the previous example is in error in one respect. If
Stocks do poorly in the first year, they are likely to do somewhat poorer than initially
projected in the second year. Specifically, the *true* payment matrix is. **QQ**:

B0 S0 Bg Sg Bb Sb g 1.05 1.26 -1.00 -1.00 0 0 b 1.05 0.96 0 0 -1.00 -1.00 gg 0 0 1.05 1.26 0 0 gb 0 0 1.05 0.96 0 0 bg 0 0 0 0 1.05 1.20 bb 0 0 0 0 1.05 0.90

The correct strategy would have been **nn = inv(QQ)*c**:

B0 0.4450 S0 0.6093 Bg 0.2670 Sg 0.9680 Bb 0.3535 Sb 0.6987

which would have cost $1.0543.

The strategy adopted, using the wrong model, cost less ($1.0277), but would in fact
provide a different set of payments (**QQ*n**) from that desired:

g 0.0000 b 0.0000 gg 1.5000 gb 1.2096 bg 1.1677 bb 0.9581

If the first year turns out to be Bad, problems lie ahead. The Investment Firm
will think that it is fully hedged but wake up to find that it either owes $1.2096 with
only $1.1677 of assets (state **bg**) or that it owes $1.0000 with only
$0.9581 of assets (state **bb**).

The second example is different in kind but similar in outcome. Assume that the tree and payment matrix are completely accurate, but that the market "moves too fast" to make any trades at the end of the first period. Instead, the positions established at the outset must be held until the end of the second period. This is not unlike the experience in a number of stock markets on the day in October, 1987 known as "Black Monday", when some of the participants found that their assumption that trades could be made after relatively small price changes was in error.

In this case, the results would be as follows:

Bonds Stocks Total gg 0.2853*1.05*1.05 0.7423*1.26*1.26 1.4931 gb 0.2853*1.05*1.05 0.7423*1.26*0.96 1.2125 bg 0.2853*1.05*1.05 0.7423*0.96*1.26 1.2125 bb 0.2853*1.05*1.05 0.7423*0.96*0.96 0.9987

In this case, the Investment Firm actually makes money if the Stock reverses its
behavior (state **gb** or **bg**) but is in trouble (may not be
able to make its payments) if the stock price continues in the same direction (state **gg**
or **bb**).

Model risk is an important element whenever a dynamic strategy is adopted to provide a
desired set of cash flows. Both firms that plan to hedge obligations and those who are
counterparties for such firms must be keenly aware of the possibility that the underlying
model is wrong in some sense or another. A *stress test*, in which changes in the
underlying model are examined to estimate the magnitudes of likely deviation, can prove
valuable in assessing the degree of the danger associated with this type of risk.

Thus far, our examples have involved a specified set of cash flows at each of the nodes
in the time-state tree. Futures and forward contracts have such attributes, as do
many swap agreements. However, a great many financial arrangements involve one or
more *options*: at certain times and under some or all conditions, one or both
parties may change the pattern of remaining cash flows.

Consider first a *European Call Option* which allows the option holder (buyer)
to "call away" a security or stock index for a pre-specified amount at a given
date. To illustrate, we use the previous example in which a Stock price can increase to
1.26 times its prior value or decrease to 0.96 times its prior value in each of two
periods while a Bond grows to 1.05 times its prior value in each period. We wish to
analyze an option to call away the stock for $1.10 at the end of the second period.

The value of the Stock at the end of that period will depend on the final state of the world, as follows:

gg $1.5876 gb $1.2096 bg $1.2096 bb $0.9216

Clearly, it would be foolish to pay $1.10 for something worth less. Hence *optimal
exercise* involves choosing to let the option expire in state **bb** and
exercising it in every other state. The net value received in each state will thus
be:

gg $0.4876 gb $0.1096 bg $0.1096 bb $0

In terms of the full vector of possible cash flows, **c**:

g 0 b 0 gg 0.4876 gb 0.1096 bg 0.1096 bb 0

The cost of providing these flows with a dynamic strategy equals **p*c**:

$ 0.0816

The replicating portfolio (strategy) is **n**:

B0 -0.5220 S0 0.6036 Bg -1.0476 Sg 1.2600 Bb -0.3340 Sb 0.3653

The initial position involves the investment of $0.0816 of the investor's money plus
$0.5220 of borrowed funds to purchase $0.6036 of the Stock. Subsequently, the
positions are adjusted, depending on the state of the world, but in each case the strategy
combines borrowing (a short position in the Bond) with investment (a long position in the
Stock).

A European option may be exercised only on its *expiration date* . An *American
option* may be exercised at any date up to and including its expiration date. Analysis
of the latter is somewhat more complex than that of the former.

Consider an American *put option* that allows the holder to "put"
(sell) the Stock to the option writer (seller) at a price of $1.20 at either time period 1
or time period 2. If the option is held until time period 2, it should be left to expire
worthless in all but state **bb**. In this case, the option will be worth
$0.2784 since it can be used to sell a stock worth only $0.9216 for $1.20.

The figure below shows the situation diagramatically. The values in the boxes for time period 2 indicate the cash flows if the option is held until time period 2 and then exercised optimally.

Should the option be exercised at the end of period 1? Consider first the situation if the first year is Good. The Stock will be worth $1.26. If the option were exercised, the holder would sell something worth $1.26 for $1.20, thereby losing $0.06. Moreover, the game would be over. It is immediately apparent that it is better to continue (to get zero) than to exercise and obtain $ -0.06.

The situation at the end of a Bad year 1 is not as clear. Since the Stock will be worth
$0.96, immediate exercise will net $0.24, as shown in the diagram. Is it better to take
this amount or to continue to hold the option in the hope of receiving either 0 (state **bg**)
or 0.2784 (state **bb**) at the end of the next year?

The question can be posed in terms of alternative vectors of cash flows. Which is
better? **c1**:

g 0 b 0 gg 0 gb 0 bg 0 bb 0.2784

or **c2**:

g 0 b 0.24 gg 0 gb 0 bg 0 bb 0

The answer is easily found by pricing the two alternatives:

p*c1 = 0.1237 p*c2 = 0.1600

Clearly, it is better to exercise the put at the end of year 1 if the stock price
falls. Planning to do so makes the option worth $0.1600 at the outset. Planning to
not do so makes it worth only $0.1237.

The figure below shows the tree after it has been "pruned" to include only optimal paths.

The procedure for pruning is simple conceptually. First, the tree is *priced*,
using the standard securities, so that the value of a payment at each node is known. Then
one starts at the end, working back one node at a time. The present value of the cash
flows associated with one decision (here, to exercise the option) is compared with the
present value of the cash flows associated with the alternative decision (here, to not
exercise the option). The better choice is retained and the poorer one discarded. The
process is performed first for all the nodes at the last time period. Then it is performed
for all the nodes at the penultimate (next-to-last) period, then for the period before
that, and so on. To speed up the process, each node can be assigned a present value based
on optimal choices at subsequent nodes. The set of such values for the nodes at time
period **t** can then be used when evaluating choices for nodes at time
period **t-1**. The final result will be a set of rules for making optimal
choices, a corresponding set of cash flows, and the associated present value which will be
the largest possible, given the alternatives.

If a contract between party A and party B gives B one or more options, how should party
A arrive at an appropriate price? One might assume that party B will act optimally and
hence follow the procedure described above. However, in many cases option-holders do not
do this. For example, homeowners who borrow money via *mortgages* often retain an
option to prepay their loan at fixed amounts, regardless of the course of interest rates.
Pools of such mortgages are frequently assembled and sold as an Investment Product. The
value of such a pool will depend critically on the nature of *prepayments* by the
individual mortgagees. Consider a borrower who has a $100 8% loan with one year to run.
She can either pay $108 in a year or $100 today. If the current rate of interest is 7%, it
is to her advantage to "pay off" the loan for $100 with money borrowed at 7%,
thus replacing an obligation to pay $108 with one to pay $107. If, on the other hand,
interest rates are 9%, it would be undesirable for her to pay off the loan.

In practice, a mortgagee may fail to prepay a loan when interest rates fall below the
rate at which the mortgage was issued, due to costs or inattention. Moreover, some will
pay off loans when interest rates are above the initial rate, due to a need to sell a
house, etc.. The prices of mortgage pool securities are typically higher than they would
be if borrowers always exercised optimally in a narrow sense. Those who analyze such
products incorporate a *prepayment model* in their calculations, based on observed
behavior of a class of borrowers. Profits can be made by analysts who utilize a model
superior to that reflected in market prices. On the other hand, losses can be incurred by
those with inferior models. To a major extent, the competition among active managers of
funds that utilize mortgage instruments is a competition among prepayment models.

When both parties to an agreement retain options, valuation requires assumptions about
the behavior of each one. Thus a *convertible callable bond* provides the issuer
with an option to call the bond from the holder under certain conditions and an option for
the holder to convert the bond into the issuer's stock under some conditions. If each
party is assumed to exercise optimally, valuation can proceed using the general procedure
outlined above. Otherwise, more complex assumptions are required.

Whatever the model used to predict choices made when options are available, the goal is to reduce the problem to one involving a vector of time and state-contingent cash flows. If markets for the associated securities are sufficiently complete, the value of an Investment Product with options and a replicating dynamic strategy can be determined.

The instruments that we have examined in the last few examples are all *derivatives*
-- Investment Products whose value depends on the values of one or more *underlying
securities*. We have considered only cases in which a derivative is tied
to one security value. Such instances are well suited to binomial models of the
behavior of the value of the underlying security. More complex derivatives may be
based on the behavior of the prices of two or more securities or on values of
non-investment vehicles (e.g. the average temperature in July at a particular resort).
The farther the underlying value from that of a traded security, the less likely it
is that a replicating portfolio can be determined and the derivative's value established
definitively. In such cases perfect hedging is impossible and the specter of
counterparty risk looms especially large.

It is an overstatement to say that an Investor can attain *any* desired pattern
of time and state-contingent cash flows via either Investment Products or dynamic
strategies. However, the range of possibilities is very large indeed and growing larger by
the day. This leaves the Analyst with two key questions: what set of payments is the most
desirable and what is the best way to achieve the desired outcome?

If an Investment Product is utilized, there is the danger that the counterparty will
fail to make all required payments, at least in some circumstances. The more exotic the
derivative, the greater such *counterparty risk* is likely to be. On the other
hand, if the Investor undertakes a dynamic strategy, he or she is directly (rather than
indirectly) subject to *model risk*. An Investor who chooses a derivative
Investment Product will need to examine the assets and other liabilities of the
counterparty. One who chooses a dynamic strategy will have to examine the credentials and
methods utilized by his or her Analyst.