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Presentation Speech - The Sveriges Riksbank (Bank
of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel
KUNGL. VETENSKAPSAKADEMIEN THE ROYAL SWEDISH ACADEMY OF SCIENCES
14 October 1997
The
Royal Swedish Academy of Sciences has decided to award the
Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel,
1997, to
Professor Robert C. Merton, Harvard University, Cambridge,
USA and
Professor Myron S. Scholes, Stanford University, Stanford,
USA
for a new method to determine the value of derivatives.
Robert C. Merton and Myron S. Scholes have, in collaboration with
the late Fischer Black, developed a pioneering formula for the
valuation of stock options. Their methodology has paved the way for
economic valuations in many areas. It has also generated new types
of financial instruments and facilitated more efficient risk
management in society.
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In a modern market
economy it is essential that firms and households are able to select
an appropriate level of risk in their transactions. This takes place
on financial markets which redistribute risks towards those agents
who are willing and able to assume them. Markets for options and
other so-called derivatives are important in the sense that agents
who anticipate future revenues or payments can ensure a profit above
a certain level or insure themselves against a loss above a certain
level. (Due to their design, options allow for hedging against
one-sided risk - options give the right, but not the obligation, to
buy or sell a certain security in the future at a prespecified
price.) A prerequisite for efficient management of risk, however, is
that such instruments are correctly valued, or priced. A new method
to determine the value of derivatives stands out among the foremost
contributions to economic sciences over the last 25 years.
This year´s laureates, Robert Merton and Myron
Scholes, developed this method in close collaboration with
Fischer Black, who died in his mid-fifties in 1995. These three
scholars worked on the same problem: option valuation. In 1973,
Black and Scholes published what has come to be known as the Black-Scholes
formula. Thousands of traders and investors now use this formula
every day to value stock options in markets throughout the world.
Robert Merton devised another method to derive the formula that
turned out to have very wide applicability; he also generalized the
formula in many directions.
Black, Merton and Scholes thus laid the foundation for the rapid
growth of markets for derivatives in the last ten years. Their
method has more general applicability, however, and has created new
areas of research - inside as well as outside of financial
economics. A similar method may be used to value insurance contracts
and guarantees, or the flexibility of physical investment projects.
The problem
Attempts to value
derivatives have a long history. As far back as 1900, the French
mathematician Louis Bachelier reported one of the earliest attempts
in his doctoral dissertation, although the formula he derived was
flawed in several ways. Subsequent researchers handled the movements
of stock prices and interest rates more successfully. But all of
these attempts suffered from the same fundamental shortcoming: risk
premia were not dealt with in a correct way.
The value of an option to buy or sell a share depends on the
uncertain development of the share price to the date of maturity. It
is therefore natural to suppose - as did earlier researchers - that
valuation of an option requires taking a stance on which risk
premium to use, in the same way as one has to determine which risk
premium to use when calculating present values in the evaluation of
a future physical investment project with uncertain returns.
Assigning a risk premium is difficult, however, in that the correct
risk premium depends on the investor´s attitude towards risk.
Whereas the attitude towards risk can be strictly defined in theory,
it is hard or impossible to observe in reality.
The method
Black, Merton and
Scholes made a vital contribution by showing that it is in fact not
necessary to use any risk premium when valuing an option. This does
not mean that the risk premium disappears; instead it is already
included in the stock price.
The idea behind their valuation method can be illustrated as
follows:
Consider a so-called European call option that gives the right to
buy one share in a certain firm at a strike price of $ 50, three
months from now. The value of this option obviously depends not only
on the strike price, but also on today´s stock price: the
higher the stock price today, the greater the probability that it
will exceed $ 50 in three months, in which case it pays to exercise
the option. As a simple example, let us assume that if the stock
price goes up by $ 2 today, the option goes up by $ 1. Assume also
that an investor owns a number of shares in the firm in question and
wants to lower the risk of changes in the stock price. He can
actually eliminate that risk completely, by selling (writing) two
options for every share that he owns. Since the portfolio thus
created is risk-free, the capital he has invested must pay exactly
the same return as the risk-free market interest rate on a
three-month treasury bill. If this were not the case, arbitrage
trading would begin to eliminate the possibility of making a
risk-free profit. As the time to maturity approaches, however, and
the stock price changes, the relation between the option price and
the share price also changes. Therefore, to maintain a risk-free
option-stock portfolio, the investor has to make gradual changes in
its composition.
One can use this argument, along with some technical assumptions, to
write down a partial differential equation. The solution to this
equation is precisely the Black-Scholes´ formula. Valuation of
other derivative securities proceeds along similar lines.
The Black-Scholes formula
Black and Scholes´ formula for a European call option can be
written as
where the variable d is defined by
According to this formula, the value of the call option C, is
given by the difference between the expected share value - the first
term on the right-hand side - and the expected cost - the second
term - if the option right is exercised at maturity. The formula
says that the option value is higher the higher the share price
today S, the higher the volatility of the share price
(measured by its standard deviation) sigma, the higher the risk-free
interest rate r, the longer the time to maturity t,
the lower the strike price L, and the higher the probability
that the option will be exercised (the probability is evaluated by
the normal distribution function N ).
Other applications
Black,
Merton and Scholes´ method has become indispensable in the
analysis of many economic problems. Derivative securities constitute
a special case of so-called contingent claims and the valuation
method can often be used for this wider class of contracts. The
value of the stock, preferred shares, loans, and other debt
instruments in a firm depends on the overall value of the firm in
essentially the same way as the value of a stock option depends on
the price of the underlying stock. The laureates had already
observed this in their articles published in 1973, thereby laying
the foundation for a unified theory of the valuation of corporate
liabilities.
A guarantee gives the right, but not the obligation, to exploit it
under certain circumstances. Anyone who buys or is given a guarantee
thus holds a kind of option. The same is true of an insurance
contract. The method developed by this year´s laureates can
therefore be used to value guarantees and insurance contracts. One
can thus view insurance companies and the option market as
competitors.
Investment decisions constitute another application. Many
investments in equipment can be designed to allow more or less
flexibility in their utilization. Examples include the ease with
which one can close down and reopen production (in a mine, for
instance, if the metal price is low) or the ease with which one can
switch between different sources of energy (if, for instance, the
relative price of oil and electricity changes). Flexibility can be
viewed as an option. To choose the best investment, it is therefore
essential to value flexibility in a correct way. The Black-Merton-Scholes´
methodology has made this feasible in many cases.
Banks and investment banks regularly use the laureates´
methodology to value new financial instruments and to offer
instruments tailored to their customers´ specific risks. At
the same time such institutions can reduce their own risk exposure
in financial markets.
Other
research contributions
Besides
their valuation method, Merton and Scholes have made several
significant contributions to financial economics. Merton has
developed a new powerful method for analyzing consumption and
investment decisions over time, and generalized the so-called CAPM
(the valuation model for which William Sharpe was awarded the Prize
in 1990) from a static to a dynamic setting. Scholes has clarified
the impact of dividends on stock market values, together with Black
and Miller (Merton Miller was awarded the Prize in 1990 for his
contributions to corporate finance), and made empirical
contributions, for example concerning estimation of the so-called
beta value (a risk measure in the CAPM).
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Further
reading
Additional
background material on the Bank of Sweden Prize in Economic Sciences
in Memory of Alfred Nobel 1997, The Royal Swedish Academy of
Sciences
Black, F. och M. Scholes, 1973, "The Pricing of Options and
Corporate Liabilities", Journal of Political Economy,
Vol. 81, pp. 637-654.
Black, F., 1989, "How We came Up with the Option Formula",
The Journal of Portfolio Management, Vol. 15, pp. 4-8
Hull, J.C., 1997, Options, Futures and Other Derivates, 3rd
edition, Prentice Hall
Merton, R.C., 1973, "Theory of Rational Option Pricing", Bell
Journal of Economics and Management Science, Vol. 4, pp.
141-183.
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Robert
C. Merton, was born in 1944 in New York, USA. He received his
Ph.D. in Economics in 1970 at MIT, Cambridge, USA. He currently
holds the George Fisher Baker Professorship in Business
Administration at Harvard Business School, Boston, USA.
Professor Robert C. Merton
Graduate School of Business Administration
Morgan Hall, Soldiers Field
Boston, MA 02163, USA
Myron S. Scholes, was born in 1941. He received his Ph.D. in
1969 at University of Chicago, USA. He currently holds the Frank E.
Buck Professorship of Finance at the Graduate School of Business and
is Senior Research Fellow at the Hoover Institution at Stanford
University, Stanford, USA
Professor Myron S. Scholes
Graduate School of Business
Stanford University
Stanford, CA 94305, USA
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