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 prices ps:
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:
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 portfolio n = 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.
