Full text of Economic Commentary (Federal Reserve Bank of Cleveland) : Options and the Future : What Do Markets Think?, October 01, 2002

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October 1, 2002

Federal Reserve Bank of Cleveland

Options and the Future: What Do
Markets Think?
by Ben Craig

ach business day, and particularly
after major events, millions of people
closely read reports of the Dow Jones
Industrial Average, the S&P 500, longterm interest rates, and the value of the
dollar. These people are not just checking the value of their portfolio or scanning interest rates before buying a
house. Rather, they know these prices
contain valuable information that can
help them gauge the impact of current
events or monetary policy on the economy as a whole. And the prices of stocks
and bonds have a fair claim to provide
this information—they are the amalgamation of buy and sell decisions of
many investors, each putting their
money on the line. The information in
asset prices is often surprisingly correct:
using information in frozen orange juice
futures prices, Richard Roll predicted
central Florida weather better than published meteorological forecasts.
What people may not realize, however, is
that market prices can yield predictions
not only about the timing or direction of
future events, but also about the markets’
degree of certainty about events transpiring as predicted. Is the market offering
equal chances of good or bad times, or is
it clearly predicting an anemic recovery?
While the picture of two old stockjobbers
standing around lamenting that the “markets are nervous” is a cliché, the increasing depth and sophistication of financial
markets over the past 25 years allow
economists to make this notion of market
certainty—or as economists refer to it,
uncertainty—more precise. More specifically, the tremendous growth in
options—contracts that give a party the

ISSN 0428-1276

right but not the obligation to buy (or sell)
at a given price—make it possible to
extract more detailed information from
market prices. This Commentary explores
the information extractable from option
prices and considers how policymakers
might use this information.
■

Options Markets

A typical options contract offers a person a payment based on the value of an
underlying good, security, or index
(called the underlying) at a future date
(the expiration date). A call option is an
option to buy the underlying at the expiration date for a certain price, called the
strike price. A put option is an option to
sell the underlying at the expiration date
at the strike price. Table 1 illustrates the
payoffs to the holder of a call or put
option on a stock with the strike price of
$55.1 Unlike holders of the underlying
stock, option holders exercise their
option only when it is in their interest, so
their loss is limited to the price paid for
the option. The call option lets you gain
from stock appreciation without losing if
stocks fall. The put lets you gain from
falling stock prices without losing if
stocks rise.
■

Uncovering Uncertainty

How can options tell us about the markets’ view of uncertainty? To see how,
we need to be more precise about what
we mean by uncertainty. For this, it pays
to draw an analogy to gambling. Consider a simple dice game where you pay
some price to play and get $1 if a six
turns up. The price of playing the game
is determined by a group of “investors.”
To make things interesting, the die may

We’re used to hearing analysts make
predictions about where the economy
is headed based on changes in the
prices people are paying for stocks,
futures, or other assets. Now, recent
research is showing how we can
analyze the prices of sophisticated
new investment products, like
options, to also gauge the probability
assigned by the markets to possible
future events. In short, we can calculate how likely market participants
feel it is that an event will take place
in the future.
be crooked, so you don’t know the odds.
The investors don’t know whether the
die is crooked either, but they have spent
a long time thinking about this particular
game with this particular die.
In this hypothetical game, one can tell
much about the probabilities of the die’s
crookedness from the price the investors
set for playing the game. If the price is
17 cents (or about $1 divided by six),
you might conclude that the die is probably fair. If, on the other hand, the price is
60 cents, you might conclude that the
probability a six will turn up is much
higher, probably greater than half.
The price of playing will equal the
chance that a six will turn up (on average, of course). However, if you had a
lot of investors offering different prices
for other numbers, you could also get the
odds on any other number turning up,
which in turn would provide you with
a measure of the uncertainty in this

“market.” That is, you get what is called
the probability distribution for the set of
all numbers on the die. This is not just
some summary of the odds—the mean
or the variance or what have you—
but rather the chances for each separate
event (each number of the die turning
up, in this case). This is important
because most summary measures,
such as means, leave important
information out.
Can you do the same thing for the stock
market, for example, find the odds that
the Dow Jones Industrial Average will
go up by 5, 10, or 13 points in one day?
Using options, the answer is yes. The
key in the dice example was having a
simple contract that paid off only if a six
came up. Options can be combined in
ways that allow an investor to isolate
risks associated with particular events
in the same way. Next, I show how a
combination of long and short positions
in the options markets (that is, offers
to buy or sell a contract) can extract
information related to a single event
in the future.
To isolate risks in the stock market, all
that is needed is to construct a portfolio
that has a payoff of $1 if the price of the
underlying stock is a single arbitrary
price (say $55) at the expiration date.
This portfolio would hold two call
options that the owner has purchased,
one with a strike price of $54 and one
with a strike price of $56, and two calls
that the owner has sold (referred to as
write calls), each with a strike price of
$55. As shown in table 2, this portfolio
pays $1 if the underlying price is $55 on
the expiration date. Using options at
different strike prices, you can construct
a portfolio that incorporates the information about where the stock price is
likely to be on the expiration date. (That
is, you get the probability distribution of
stock prices on a given day.) Incidentally, you can do the same thing with a
comparable portfolio of put options.2
Indeed, this distribution contains far
more information than just a single
number that measures the degree of
uncertainty, such as the variance from
statistics theory. To continue our analogy with the crooked dice, suppose that
two dice were thrown in a game of
craps. Numbers such as those adding up
to seven, eleven, or two are of special
importance, so it could be helpful to
know the probability associated with

TABLE 1: PAYOFF TO OPTIONS: STRIKE PRICE OF $55

Price at expiration

Futures contract
(that costs $55)

50
51
52
53
54
55
56
57
58
59
60

Call option

–5
–4
–3
–2
–1
0
1
2
3
4
5

0
0
0
0
0
0
1
2
3
4
5

Put option
5
4
3
2
1
0
0
0
0
0
0

TABLE 2: A PORTFOLIO THAT PAYS $1 AT SELECTED
EXPIRATION PRICE (HERE $55)

Strike price

Buy 1
$54

Price at expiration
50
51
52
53
54
55
56
57
58

Holdings
Write 2
$55

Total payoff
of portfolio

Payoff of each call
0
0
0
0
0
1
2
3
4

specific numbers. In the same way, there
are times when investors or policymakers
need to know the market assessment of
the probability of specific events. For
example, a construction firm whose
break-even point is at a given interest rate
may be interested in the probability that
interest rates will be higher than this rate.
■

Buy 1
$56

An Example

While financial engineers and corporate
risk managers would obviously be interested in the distribution of prices, the
distribution can also be of interest to
others. For example, looking at how the
distribution of stock prices changes after
an action by the Federal Reserve’s Federal Open Market Committee (FOMC) is
much richer than statements such as “the
markets fell as a result of the monetary
policy announcement.” Using options on
the S&P 500, a policy announcement can
be analyzed for its effect on the markets’
assessment of future price changes.
A priori, one could argue that an

0
0
0
0
0
0
–2
–4
–6

0
0
0
0
0
0
0
1
2

0
0
0
0
0
1
0
0
0

announcement causes more information
to come to the market, so the FOMC
announcement gives more certainty to
investors. On the other hand, the FOMC
announcement could indicate a change
in policy, perhaps injecting more uncertainty into the market.
Note that there is always a problem
with assuming that because an action
occurred and market prices changed
afterward, the markets must have
reacted to that action. In other words,
it may be difficult to sort out cause
and effect. The most we can say with
certainty is that the distribution of stock
market outcomes changed shortly after
a monetary announcement was made.
However, the probability distribution
will give an indication of how the
markets forecast change following a
policy announcement.
For example, on April 18, 2001, between
regular meetings, the FOMC decided to

FIGURE 1: PROBABILITIES FOR CHANGES IN S&P 500

Probabilities
0.2
0.18
0.16
S&P 500 4/17/01
0.14

S&P 500 4/19/01

0.12
0.1
0.08
0.06
0.04
0.02
0
< –225 –225 –200 –175 –150 –125 –100 –75 –50 –25 0 25 50 75 100 125 150 175 200 225 >225
Deviations from futures value

SOURCE: Author’s calculations.

cut the overnight interest rate by 50 basis
points. The statement issued with this
action said: “The Committee continues to
believe that against the background of its
long-run goals of price stability and sustainable economic growth and of the
information currently available, the risks
are weighted mainly toward conditions
that may generate economic weakness in
the foreseeable future.” Newspaper
accounts the next day said the markets
did not anticipate the amount of the cut
and reacted strongly as a result. Equity
markets gained in value: the Nasdaq the
day after the cut was 9.8 percent above
its price of the previous day, and the Dow
Jones gained almost 5 percent. The general impression was that the equity markets responded to the announcement by
increasing firms’ expected earnings.
The FOMC’s press release itself, however, hints that investors’ reaction was in
fact more complicated. The comment
about the “rising uncertainty about the
business outlook” suggests that the markets might have become less certain
about future earnings. On the other hand,
perhaps those earnings were uncertain
because of uncertainty about FOMC
actions. In light of the press release
accompanying the action, it would be
interesting to know whether equitymarket uncertainty diminished or rose as
a result of the policy action.

If we look at figure 1, we see the
probabilities for various values of the
S&P 500 as calculated from options
for the June 2001 expiration date that
was about two months away. These are
calculated for the day before and the
day following the interest rate
announcement. The futures price for
the index (for June delivery) rose
about 5 percent, from 1195.5 to
1253.7, during these two days. On the
horizontal axis is the value in points
that the index may be expected to rise
above the futures value, in 25-point
increments. The probabilities are calculated for the event that on the June
21 expiration date, the index falls
between the value of the horizontal
axis and 25 points below it. Thus, the
label “–25” corresponds to the event
where, by June 21, the index falls
between 50 and 25 points below the
futures value, “25” corresponds to a
rise between 0 and 25 points above the
futures value, and so forth. The probabilities are calculated, approximately,
around where investors expect, on
average, the market to be for each day,
so that a comparison between the two
days can be made.
What these probabilities show is that the
uncertainty surrounding the possible
future values of the index did indeed
diminish from the day before the action
to the day afterward. In other words, the

probability bars for April 19 are higher
around the center of the distribution than
are the corresponding bars for the earlier
date, which are generally higher in the
ends, indicating that on April 17, there
was a greater belief that changes in the
index might be extreme. Further, much
of the probability on April 19 was
assigned to the possibility that the market would advance normally and the
index would gain between 25 and 50
points, as indicated by the long red bar at
50. Thus, the FOMC action was associated with a change in market beliefs,
which became more concentrated
around the futures contract value, that is,
near the value that would give investors
an average rate of return.
These results must be taken with a
grain of salt. Measurement error can be
quite large when working with option
prices that are set by trades throughout
the day. Even so, they provide some
evidence that the FOMC action reduced
market uncertainty, at least for the day
following the announcement of a 50basis-point decrease.
■

Conclusion

Investing, because it involves predicting
the future, is inherently risky. Yet that
means asset prices, the outcome of
many traders doing their best to forecast the future, can give us information
about the risk of today’s economy. This
can also tell us how policy or other
events change that risk. Recent research
has concentrated on reducing the error
in calculating these market measurements of the distribution and with it
their ability to forecast the future. This
is a fruitful and active area of research
that should give policymakers an additional piece of immediate information
with which to evaluate the effects of
their actions.
■

Footnotes

1. The payoffs shown in table 1 do not
include several things that must be
taken into account by a person buying
the option. First, the expiration date is
in the future, so the payoffs are paid at
the expiration date. This means that the
payoff must be discounted back to
“current dollars” when evaluating the
value of an options portfolio. Second,
an investor, when calculating the possible profits from purchasing an option,
must also subtract the price of the
option from its expected benefits.

2. Call and put options may be American or European contracts. The holder
of an American option can exercise
it any time up to and including the
expiration date. European options can
be exercised only on the expiration
date (see Cox and Rubinstein in the
recommended reading). This Commentary assumes that the options are
European options and that investors
are risk neutral.
■

Full text of Economic Commentary (Federal Reserve Bank of Cleveland) : Options and the Future : What Do Markets Think?, October 01, 2002

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