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, and

Professor **Myron S. Scholes**, Stanford University

*for a new method to determine the value of derivatives.*

Risk management is essential in a modern
market economy. Financial markets enable firms and households to
select an appropriate level of risk in their transactions, by
redistributing risks towards other agents who are willing and
able to assume them. Markets for options, futures and other
so-called derivative securities - derivatives, for short - have a
particular status. Futures allow agents to hedge against upcoming
risks; such contracts promise future delivery of a certain item
at a certain price. As an example, a firm might decide to engage
in copper mining after determining that the metal to be extracted
can be sold in advance at the futures market for copper. The risk
of future movements in the copper price is thereby transferred
from the owner of the mine to the buyer of the contract. Due to
their design, options allow agents to hedge against one-sided
risks; options give the right, but not the obligation, to buy or
sell something at a pre-specified price in the future. An
importing British firm that anticipates making a large payment in
US dollars can hedge against the one-sided risk of large losses
due to a future depreciation of sterling by buying call options
for dollars on the market for foreign currency options. Effective
risk management requires that such instruments be correctly
priced.

Fischer Black, Robert Merton and Myron Scholes made a pioneering
contribution to economic sciences by developing a new method of
determining the value of derivatives. Their innovative work in
the early 1970s, which solved a longstanding problem in financial
economics, has provided us with completely new ways of dealing
with financial risk, both in theory and in practice. Their method
has contributed substantially to the rapid growth of markets for
derivatives in the last two decades. Fischer Black died in his
early fifties in August 1995.

Black, Merton and Scholes´ contribution extends far beyond
the pricing of derivatives, however. Whereas most existing
options are financial, a number of economic contracts and
decisions can also be viewed as options: an investment in
buildings and machinery may provide opportunities (options) to
expand into new markets in the future. Their methodology has
proven general enough for a wide range of applications. It can
thus be used to value not only the flexibility of physical
investment projects but also insurance contracts and guarantees.
It has created new areas of research - inside as well as outside
of financial economics.

Attempts to value options and other
derivatives have a long history. One of the earliest endeavors to
determine the value of stock options was made by Louis Bachelier
in his Ph.D. thesis at the Sorbonne in 1900. The formula that he
derived, however, was based on unrealistic assumptions, a zero
interest rate, and a process that allowed for a negative share
price.

Case Sprenkle, James Boness and Paul Samuelson improved on
Bachelierís formula. They assumed that stock prices are
log-normally distributed (which guarantees that share prices are
positive) and allowed for a non-zero interest rate. They also
assumed that investors are risk averse and demand a risk premium
in addition to the risk-free interest rate. In 1964, Boness
suggested a formula that came close to the Black-Scholes formula,
but still relied on an unknown interest rate, which included
compensation for the risk associated with the stock.

The attempts at valuation before 1973 basically determined the
expected value of a stock option at expiration and then
discounted its value back to the time of evaluation. Such an
approach requires taking a stance on which risk premium to use in
the discounting. This is because the value of an option depends
on the risky path of the stock price, from the valuation date to
maturity. But assigning a risk premium is not straightforward.
The risk premium should reflect not only the risk for changes in
the stock price, but also the investorsí attitude towards
risk. And while the latter can be strictly defined in theory, it
is hard or impossible to observe in reality.

This yearís laureates resolved these
problems by recognizing that it is not necessary to use any risk
premium when valuing an option. This does not mean that the risk
premium disappears, but that it is already incorporated in the
stock price. In 1973 Fischer Black and Myron S. Scholes published
the famous option pricing formula that now bears their name
(Black and Scholes (1973)). They worked in close cooperation with
Robert C. Merton, who, that same year, published an article which
also included the formula and various extensions (Merton
(1973)).

The idea behind the new method developed by Black, Merton and
Scholes can be explained in the following simplified way.
Consider a so-called European call option that gives the right to
buy a certain share at a strike price of $100 in three months. (A
European option gives the right to buy or sell only at a certain
date, whereas a so-called American option gives the same right at
any point in time up to a certain date.) Clearly, the value of
this call option depends on the current share price; the higher
the share price today the greater the probability that it will
exceed $100 in three months, in which case it will pay to
exercise the option. A formula for option valuation should thus
determine exactly how the value of the option depends on the
current share price. How much the value of the option is altered
by a change in the current share price is called the "delta" of
the option.

Assume that the value of the option increases by $1 when the
current share price goes up $2 and decreases by $1 when the stock
goes down $2 (i.e. delta is equal to one half). Assume also that
an investor holds a portfolio of the underlying stock and wants
to hedge against the risk of changes in the share price. He can
then, in fact, construct a risk-free portfolio by selling
(writing) twice as many options as the number of shares he owns.
For reasonably small increases in the share price, the profit the
investor makes on the shares will be the same as the loss he
incurs on the options, and vice versa for decreases in the share
price. As the portfolio thus constructed is risk free, it must
yield exactly the same return as a risk-free three-month treasury
bill. If it did not, arbitrage trading would begin to eliminate
the possibility of making risk-free profits.

As the share price is altered over time and as the time to
maturity draws nearer, the delta of the option changes. In order
to maintain a risk-free stock-option portfolio, the investor has
to change its composition. Black, Merton and Scholes assumed that
such trading can take place continuously without any transaction
costs (transaction costs were later introduced by others). The
condition that the return on a risk-free stock-option portfolio
yields the risk-free rate, at each point in time, implies a
partial differential equation, the solution of which is the
Black-Scholes formula for a call option:

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 price -
the first term on the right-hand side - and the expected cost -
the second term - if the option is exercised. The option value is
higher, the higher the current share price *S*, the higher
the volatility of the share price (as 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 (this probability is, under risk
neutrality, evaluated by the standardized normal distribution
function *N* ). All the parameters in the equation can be
observed except sigma, which has to be estimated from market
data. Alternatively, if the price of the call option is known,
the formula can be used to solve for the marketís estimate
of sigma.

The option pricing formula is named after Black and Scholes
because they were the first to derive it. Black and Scholes
originally based their result on the capital asset pricing model
(CAPM, for which Sharpe was awarded the 1990 Prize). While
working on their 1973 paper, they were strongly influenced by
Merton. Black describes this in an article (Black (1989)):

"As we worked on the paper we had long discussions with Robert
Merton, who also was working on option valuation. Merton made a
number of suggestions that improved our paper. In particular, he
pointed out that if you assume continuous trading in the option
or the stock, you can maintain a relation between them that is
literally riskless. In the final version of the paper we derived
the formula that way because it seemed to be the most general
derivation."

It was thus Merton who devised the important generalization that
market equilibrium is not necessary for option valuation; it is
sufficient that there are no arbitrage opportunities. The method
described in the example above is based precisely on the absence
of arbitrage (and on stochastic calculus). It generalizes to
valuation of other types of derivatives. Mertonís 1973
article also included the Black-Scholes formula and some
generalizations; for instance, he allowed the interest rate to be
stochastic. Four years later, he also developed (Merton (1977)) a
more general method of deriving the formula which uses the fact
that options can be created synthetically by trading in the
underlying share and a risk-free bond.

The option-pricing formula was the solution of a more than seventy-year old problem. As such, this is, of course, an important scientific achievement. The main importance of Black, Merton and Scholes´ contribution, however, refers to the theoretical and practical significance of their method of analysis. It has been highly influential in solving many economic problems. The scientific importance extends to both the pricing of derivative securities and to valuation in other areas.

The method described here has been
enormously important for the pricing of derivative instruments.
The laureates initiated the rapid evolution of option pricing
that has taken place during the past two decades. Options on
instruments other than shares give rise to other formulas that
sometimes have to be solved numerically. The same method has been
used to determine the value of currency options, interest rate
options, options on futures, and so on.

Several of the original - and somewhat restrictive - assumptions
behind the initial derivation of the Black-Scholes formula have
subsequently been relaxed. In modern option pricing the interest
rate can be stochastic, the volatility of the stock price can
vary stochastically over time, the price process can include
jumps, transaction costs can be positive, and the price process
can be managed (e.g. as currencies are restricted to move inside
bands). These extensions all rely on the method of analysis that
Black, Merton and Scholes introduced.

Black, Merton and Scholes realized already
in 1973 that a share can be interpreted as an option on the whole
firm (the title of their 1973 article is "The pricing of options
and corporate liabilities"). When loans mature and the value of
the firm is lower than the nominal value of debt, the
shareholders have the right, but not the obligation, to repay the
loans. The method can thus be used for determining the value of
shares, which can be important if the shares are not traded.
Since other corporate liabilities are also derivative instruments
(whose value, too, depends on the value of the firm), they can be
valued using the same method. Black, Merton and Scholes thus laid
a satisfactory foundation for a unified theory to price all
corporate liabilities.

Financial contract theory is undergoing rapid development. The
Black-Merton-Scholes methodology has been important in recent
efforts to design optimal financial contracts, taking into
account various aspects of bankruptcy law.

In the choice between different investment alternatives, flexibility is a key factor. Machines may differ in their flexibility regarding the shut-down and start-up of production (as the market price of the product varies), the use of different sources of energy (as, say, the relative price between oil and electricity varies), etc. The Black-Merton-Scholes methodology can often be used to facilitate more informed investment decisions in such cases.

Many types of insurance contracts and guarantees can be valued using modern option-pricing theory. Assume that an insurance company wants to determine the value of an insurance contract that protects a bondholder against the risk that the company issuing the bond will go bankrupt. The value of such an insurance contract can be approximated by a put option on the bond with a strike price equal to the nominal value of the bond. If the value of the bond declines below the strike price, the holder has the right to sell the bond at that price - that is, his possible loss is limited. In practice, therefore, an insurance company not only competes with other insurance companies, but also with the option market.

Methods developed by Merton (in Merton (1977)) have been used to extend the dynamic theory of financial markets. In the 1950s, Kenneth Arrow and Gerard Debreu (both previous laureates) showed how individuals or companies can eliminate their particular risk profile if there exist as many independent securities as there are future states of the world. This might well require a large number of securities. These findings have now been extended to show that a few financial instruments are sufficient to eliminate risk, even when the number of future states is very large.

The Chicago Board Options Exchange
introduced trade in options in April 1973, one month before
publication of the option-pricing formula. By 1975, traders on
the options exchange had begun to apply the formula - using
especially programmed calculators - to price and protect their
option positions. Nowadays, thousands of traders and investors
use the formula every day to value stock options in markets
throughout the world.

Such rapid and widespread application of a theoretical result was
new to economics. It was particularly remarkable since the
mathematics used to derive the formula were not part of the
standard training of practitioners or academic economists at that
time.

The ability to use options and other derivatives to manage risks
is quite valuable. For instance, portfolio managers use put
options to reduce the risk of large declines in share prices.
Companies use options and other derivative instruments to reduce
risk. Banks and other financial institutions use the method
developed by Black, Merton and Scholes to develop and determine
the value of new products, sell tailor-made financial solutions
to their customers, as well as to reduce their own risks by
trading in financial markets.

Financial institutions employ mathematicians, economists and
computer experts who have made important contributions to applied
research in option theory. They have developed databases, new
methods of estimating the parameters needed to value options and
numerical methods to solve partial differential equations.

Merton and Scholes have made important
scientific contributions in addition to those described so
far.

Merton has made fundamental contributions (Merton (1969) and
(1971)) to the analysis of individual consumption and investment
decisions in continuous time. He presented (Merton (1973b)) an
important generalization of CAPM, extending it from a static to a
dynamic model. He also improved and generalized option pricing
formula in different directions. In particular, he derived
(Merton (1976)) a formula which allows stock price movements to
be discontinuous.

Scholes has studied the effect of dividends on share prices
(Black and Scholes (1974) and Miller and Scholes (1978) (1982)).
He has also made empirical contributions, e.g. regarding
estimation of b, the parameter that measures the risk of a share
in the CAPM (Scholes and Williams (1977)), and market efficiency
(Black, Jensen and Scholes (1972)).

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 method has had profound importance for economic valuations in many areas. It has also helped generate new financial instruments and facilitated more efficient management of risk in society.

Black, F. and Scholes, M., 1972, "The
Valuation of Option Contracts and a Test of Market Efficiency",
*Journal of Finance*, Vol. 27, pp. 399-417.

Black, F. and Scholes, M., 1973, "The Pricing of Options and
Corporate Liabilities", *Journal of Political Economy*, Vol.
81, pp. 637-654.

Black, F. and Scholes, M., 1974, "The Effects of Dividend Yield
and Dividend Policy on Common Stock Prices and Returns",
*Journal of Financial Economics*,Vol.1,pp.1-22.

Black, F., 1989, "How We Came Up with the Option Formula", *The
Journal of Portfolio Management*, Vol. 15, pp. 4-8.

Black F., Jensen, M.C. and Scholes, M., 1972, "The Capital Asset
Pricing Model: Some Empirical Tests" in Jensen, M.C., ed.,
*Studies in the Theory of Capital Markets*, Praeger.

Merton, R.C., 1969, "Lifetime Portfolio Selection under
Uncertainty: The Continuous-Time Case", *Review of Economics
and Statistics*, Vol. 51, pp. 247-257.

Merton, R.C., 1971, "Optimum Consumption and Portfolio Rules in a
Continuous Time Model", *Journal of Economic Theory*, Vol.
3, pp. 373-413.

Merton, R.C., 1973a, "Theory of Rational Option Pricing", *Bell
Journal of Economics and Management Science*, Vol. 4, pp.
637-654.

Merton, R.C., 1973b, "An Intertemporal Capital Asset Pricing
Model", *Econometrica*, Vol. 41, pp. 867-887.

Merton, R.C., 1976, "Option Pricing When Underlying Stock Returns
Are Discontinuous", *Journal of Financial Economics*, Vol.
3, pp. 125-144.

Merton, R.C., 1977, "On the Pricing of Contingent Claims and the
Modigliani-Miller Theorem", *Journal of Financial
Economics*, Vol. 5, pp. 141-183.

Miller, M.H. and Scholes, M., 1978, "Dividends and Taxes",
*Journal of Financial Economics*, Vol. 6, pp. 333-364.

Miller, M.H. and Scholes, M., 1982, "Dividends and Taxes: Some
Empirical Evidence", *Journal of Political Economy*, Vol.
90, pp. 1118-1141.

Scholes, M. and Williams, J., 1977, "Estimating Betas from
Nonsynchronous Data", *Journal of Financial Economics*, Vol.
5, pp. 309-327.

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