Against the Gods: The Remarkable Story of Risk (60 page)
Read Against the Gods: The Remarkable Story of Risk Online
Authors: Peter L. Bernstein
The Confederate States were the sellers of these options: they took on uncertain liabilities because they had no choice in the matter. A promise to repay the loan in Confederate dollars would have been laughed out of the credit markets or would have necessitated an intolerable double-digit interest rate. The premium the Confederates received in return from the lenders who acquired these options was a reduction in the interest rate on the loan: 7% was only about a percentage point more than the U.S. government was paying for longterm money at that time. The introduction of the options made this a transaction in which uncertainty itself was an integral part.
The history of these bonds is interesting. The subscription books were opened in March 1863, but, in keeping with the conventions of the times, the proceeds were not to be received by September. The bonds sold above their offering price for a brief period after the March offering, but then the price broke sharply as stories began to circulate about Jefferson Davis's connection with some repudiated bonds in Mississippi. Concerned that subscribers would reneg on the payments
due in September, the Confederate Treasury went into the market to
support the price by buying up some C1.4 million of the C3 million
issued. The Confederates met the payments due in September 1863 and
the two semiannual payments in 1864, but that was the end. Only
about £370,000 par value was ever redeemed in cotton.
There is more than meets the eye in the design of the cotton bond, the farmer's futures contracts, the tulip options, and mortgage prepayment privileges. Most business and financial transactions are a bet in which the buyer hopes to be buying low and the seller hopes to be selling high. One side is always doomed to disappointment. Riskmanagement products are different. They exist, not necessarily because someone is seeking a profit, but because there is a demand for instruments that transfer risk from a risk-averse party to someone willing to bear risk. In the case of the cotton loan, the Confederacy took on a foreign-exchange risk and even the risk of victory itself in order to save the difference between 7% and the interest that would have been demanded without the options; it may even have received money that would not have been forthcoming under other conditions. The lenders-the buyers of the Confederate bonds-acquired options that reduced their risk sufficiently to compensate for the lower interest rate
or for the possibility that the Confederates would lose the war. By trading uncertainty, both parties were winners.
What is an option worth? How did the traders in tulip options decide
how much to pay for a call or a put, and why did those values change
over time? How did the lenders to the Confederates decide that the
options to receive payment in sterling or francs or cotton were sufficient
to hedge the risks they took in making the loans? How much extra is the
homeowner with a prepayment privilege paying the mortgage banker?
The answers to these questions may become clearer if we look at an
example of an actively traded option on a stock. On June 6, 1995, when
AT&T stock was selling at 50, there was an option outstanding on
AT&T stock that gave the owner the right to buy one share of stock at
50 1/4 until October 15, 1995. The stock was selling for less than 50
1/4-the "strike price"; if the stock remained below the strike price for
the duration of the option, the option would be worthless and its owner
would lose the entire premium paid for it. Yet that premium is all that
the buyer of the option had at risk and all that the seller of the option
could hope to gain. If AT&T stock rose above the strike price before
October 15 by an amount greater than the option premium, the option
would generate a profit. In fact, the potential profit on the option would
be limitless.
The option on AT&T stock was selling for $2.50 on June 6, 1995.
Why $2.50?
Resolving Paccioli's unfinished game of balla was kid stuff compared
to this! We can only wonder whether two quants like Pascal and Fermat
could have come up with an answer-and why they did not even try.
The Dutch tulip mania, a striking example of what happens when "oldfashioned human hunches" take over, had occurred only twenty years
before Pascal and Fermat first laid out the principles of probability theory; the memory of it must still have been vivid when they began their
historic deliberations. Perhaps they ignored the challenge of valuing an
option because the key to the puzzle is in the price of uncertainty, a
concept that seems more appropriate to our own times than it may have
seemed to theirs.
The first effort to use mathematics rather than intuition in valuing
an option was made by Louis Bachelier back in 1900. In the 1950s and
1960s, a few more people tried their hands at it, including Paul
Samuelson.
The puzzle was finally solved in the late 1960s by an odd threesome, none of whom was yet thirty years old when their collaboration
began.' Fischer Black was a physicist-mathematician with a doctorate
from Harvard who had never taken a course in economics or finance.
He soon found his scientific academic studies too abstract for his taste
and went to work at the Boston-based management consulting firm of
Arthur D. Little. Myron Scholes had a fresh Ph.D. in finance from the
Graduate School of Business at the University of Chicago, to which he
had fled to escape his family's publishing enterprise; he had just joined
the MIT faculty. Robert C. Merton, whose first published paper was
titled "The `Motionless' Motion of Swift's Flying Island," had received
a B.S. degree in mathematical engineering at Columbia but was teaching economics at MIT as an assistant to Samuelson and was as yet without a Ph.D.
Black died in 1995 at the age of 57. He was a cool man of few
words; his presidential address to the American Economic Association
in 1985 had a one-word-one-syllable title-"Noise"-and took less
than fifteen minutes to deliver. Scholes is dark, intense, and voluble.
Merton is friendly and irrepressible. All three have been brilliant innovators in finance, beyond their contribution to option theory.
The story begins in 1965, when Black made friends with a colleague named Jack Treynor; Treynor was just starting on a path that
would lead him to become a theoretical powerhouse in the field of
finance. At the time, he was studying economics on the side under the
guidance of Franco Modigliani of the MIT faculty, who would later
earn a Nobel Prize in economics. When Treynor showed Black his
early work on a model to explain how the markets trade off risk and
return, Black was fascinated. A passionate believer in free markets,
Black decided to apply Treynor's ideas to the valuation of options, and,
to help himself along, he took Treynor's advice to join a Thursday
evening finance workshop at MIT.
Three years later, Black was still staring at equations that refused to
produce an answer. Treynor's analysis of how market fluctuations influence the valuation of individual securities simply did not fit the bill. At that point, Black recalls, "Myron Scholes and I started working
together." They had met each other at the Thursday evening workshops, where Black discovered that Scholes had been frustrated in taking the same approach to the same problem. The more they worked
together over their equations, the clearer it seemed that the answer had
nothing to do with Treynor's models of risk and reward.
In the spring of 1970, Scholes told Merton about the troubles he
and Black were having. The problem appealed to Merton immediately. He soon resolved their dilemma by pointing out that they were
on the right track for reasons they themselves had failed to recognize.
The model was soon completed.
Despite its complex algebraic appearance, the basic ideas behind the
model are easy to understand. The value of an option depends on four
elements: time, prices, interest rates, and volatility. These elements
apply to puts as well as to calls; in what follows, I explain how they
work in terms of a call option, which gives the owner the right to buy
the stock at a specified price.
The first element is the period of time until the option is due to
expire; when the time to expiration is long, the option will be worth
more than when the time is short. The second element is the spread
between the current price of the stock and the price specified in the
option contract at which the owner can buy or sell the stock-this is
known as the strike price; the option will be worth more when the
actual price is above the strike price than when it is below the strike
price. Third, the value also depends on the interest the buyer can earn
on his money while waiting to exercise the option as well as the
income the seller can receive on the underlying asset over the same
time period. But what really matters is the fourth element: the expected
volatility of the underlying asset, such as the AT&T stock in the example above, where AT&T was selling for 50 and the owner of the option
had the right to buy it at 50 1/4 any time between June 6 and October
15, 1995.
The probability that the price of AT&T stock might go up-or
down-is irrelevant. The only thing that matters is how far the stock
price might move, not the direction in which it moves. The notion
that the direction of price change is irrelevant to the valuation of an
option is so counterintuitive that it explains in part why Black and
Scholes took so long to come up with the answer they were seeking even when it was right in front of them. But it unlocks the puzzle
because of the asymmetric nature of the option itself. the investor's
potential loss is limited to the premium, while the potential profit is
unlimited.
If AT&T stock goes to 45, or 40, or even to 20 during the life of
the option, the owner of the option still stands to lose no more than
$2.50. Between 50 1/4 and 52 3/4, the owner will gain less than $2.50.
Above 52 3/4, the potential profit is infinite-at least in theory. With
all the variables cranked in, the Black-Scholes model indicates that the
AT&T option was worth about $2.50 in June 1995 because investors
expected AT&T stock to vary within a range of about 10%, or five
points, in each direction during the four months the option would be
in existence.
Volatility is always the key determinant. By way of contrast to
AT&T, consider the stock of software leader Microsoft. On the same
day that AT&T stock was at 50 and its option was selling for $2.50,
Microsoft stock was selling at 83 1/8, and an option to buy a share of
Microsoft within four months at 90 was trading for $4.50. The price of
this option was 80% above the price of the AT&T option, although
Microsoft stock was selling at only about 60% above AT&T. The price
of Microsoft stock was nearly seven points away from the strike price,
compared with the mere quarter of a point difference in the case of
AT&T. The market clearly expected Microsoft to be more volatile
than AT&T. According to the Black-Scholes model, the market expected Microsoft to be exactly twice as volatile as AT&T over the following four months.
Microsoft stock is a lot riskier than AT&T stock. In 1995, AT&T had
revenues of nearly $90 billion, 2.3 million shareholders, a customer in
just about every household and every business in the nation, a weakened but still powerful monopolistic position in its industry, and a long
history of uninterrupted dividend payments. Microsoft stock had been
available to the public only since 1982, its revenues at the time were
just $6 billion, it had a much narrower customer base than AT&T, it
had brilliant competitors straining to break its hold on the software
industry, and it had never paid a dividend.
Option traders understand such differences. Anything that makes a
stock move at all is what matters, because stocks that tend to drop fast
also tend to rise fast. Buyers of options are looking for action; investors who sell options like stocks that stand still. If Microsoft goes to 100 and
the owner of the option exercises his right to "call" the stock at 90 from
the seller of the option, the seller is going to be out ten points. But if
Microsoft hangs in around 83, at which it was trading when the transaction took place, the seller of the option would walk away with the
entire premium of $4.50. By the same token, the right to prepay a
home mortgage is worth a lot more when interest rates are jumping
around than when they are stable.