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 to Professor Myron
S. Scholes, Stanford University, jointly. The prize was awarded for a
new method to determine the value of derivatives.
This sounds like a trifle achievement - but it is not. It touches upon
the very heart of the science of Economics: the concept of Risk. Risk
reflects the effect on the value of an asset where there is an option
to change it (the value) in the future.
We could be talking about a physical assets or a non-tangible asset,
such as a contract between two parties. An asset is also an investment,
an insurance policy, a bank guarantee and any other form of contingent
liability, corporate or not.
Scholes himself said that his formula is good for any situation
involving a contract whose value depends on the (uncertain) future
value of an asset.
The discipline of risk management is relatively old. As early as 200
years ago households and firms were able to defray their risk and to
maintain a level of risk acceptable to them by redistributing risks
towards other agents who were willing and able to assume them. In the
financial markets this is done by using derivative securities options,
futures and others. Futures and forwards hedge against future
(potential - all risks are potentials) risks. These are contracts which
promise a future delivery of a certain item at a certain price no later
than a given date. Firms can thus sell their future production
(agricultural produce, minerals) in advance at the futures market
specific to their goods. The risk of future price movements is
re-allocated, this way, from the producer or manufacturer to the buyer
of the contract. Options are designed to hedge against one-sided risks;
they represent the right, but not the obligation, to buy or sell
something at a pre-determined price in the future. An importer that has
to make a large payment in a foreign currency can suffer large losses
due to a future depreciation of his domestic currency. He can avoid
these losses by buying call options for the foreign currency on the
market for foreign currency options (and, obviously, pay the correct
price for them).
Fischer Black, Robert Merton and Myron Scholes developed a method of
correctly pricing derivatives. Their work in the early 1970s proposed a
solution to a crucial problem in financing theory: what is the best
(=correctly or minimally priced) way of dealing with financial risk. It
was this solution which brought about the rapid growth of markets for
derivatives in the last two decades. Fischer Black died in August 1995,
in his early fifties. Had he lived longer, he most definitely would
have shared the Nobel Prize.
Black, Merton and Scholes can be applied to a number of economic
contracts and decisions which can be construed as options. Any
investment may provide opportunities (options) to expand into new
markets in the future. Their methodology can be used to value things as
diverse as investments, insurance policies and guarantees.
Valuing Financial Options
One of the earliest efforts to determine the value of stock options was
made by Louis Bachelier in his Ph.D. thesis at the Sorbonne in 1900.
His formula was based on unrealistic assumptions such as a zero
interest rate and negative share prices.
Still, scholars like Case Sprenkle, James Boness and Paul Samuelson
used his formula. They introduced several now universally accepted
assumptions: that stock prices are normally distributed (which
guarantees that share prices are positive), a non-zero (negative or
positive) interest rate, the risk aversion of investors, the existence
of a risk premium (on top of the risk-free interest rate). In 1964,
Boness came up with a formula which was very similar to the
Black-Scholes formula. Yet, it still incorporated compensation for the
risk associated with a stock through an unknown interest rate.
Prior to 1973, people discounted (capitalized) the expected value of a
stock option at expiration. They used arbitrary risk premiums in the
discounting process. The risk premium represented the volatility of the
underlying stock.
In other words, it represented the chances to find the price of the
stock within a given range of prices on expiration. It did not
represent the investors' risk aversion, something which is impossible
to observe in reality.
The Black and Scholes Formula
The revolution brought about by Merton, Black and Scholes was
recognizing that it is not necessary to use any risk premium when
valuing an option because it is already included in the price of the
stock. In 1973 Fischer Black and Myron S. Scholes published the famous
option pricing Black and Scholes formula. Merton extended it in 1973.
The idea was simple: a formula for option valuation should determine
exactly how the value of the option depends on the current share price
(professionally called the "delta" of the option). A delta of 1 means
that a $1 increase or decrease in the price of the share is translated
to a $1 identical movement in the price of the option.
An investor that holds the share and wants to protect himself against
the changes in its price can eliminate the risk by selling (writing)
options as the number of shares he owns. If the share price increases,
the investor will make a profit on the shares which will be identical
to the losses on the options. The seller of an option incurs losses
when the share price goes up, because he has to pay money to the people
who bought it or give to them the shares at a price that is lower than
the market price - the strike price of the option. The reverse is true
for decreases in the share price. Yet, the money received by the
investor from the buyers of the options that he sold is invested.
Altogether, the investor should receive a yield equivalent to the yield
on risk free investments (for instance, treasury bills).
Changes in the share price and drawing nearer to the maturity
(expiration) date of the option changes the delta of the option. The
investor has to change the portfolio of his investments (shares, sold
options and the money received from the option buyers) to account for
this changing delta.
This is the first unrealistic assumption of Black, Merton and Scholes:
that the investor can trade continuously without any transaction costs
(though others amended the formula later).
According to their formula, the value of a call option is given by the
difference between the expected share price and the expected cost if
the option is exercised. The value of the option is higher, the higher
the current share price, the higher the volatility of the share price
(as measured by its standard deviation), the higher the risk-free
interest rate, the longer the time to maturity, the lower the strike
price, and the higher the probability that the option will be exercised.
All the parameters in the equation are observable except the volatility
, which has to be estimated from market data. If the price of the call
option is known, the formula can be used to solve for the market's
estimate of the share volatility.
Merton contributed to this revolutionary thinking by saying that to
evaluate stock options, the market does not need to be in equilibrium.
It is sufficient that no arbitrage opportunities will arise (namely,
that the market will price the share and the option correctly). So,
Merton was not afraid to include a fluctuating (stochastic) interest
rate in HIS treatment of the Black and Scholes formula.
His much more flexible approach also fitted more complex types of
options (known as synthetic options - created by buying or selling two
unrelated securities).
Theory and Practice
The Nobel laureates succeeded to solve a problem more than 70 years old.
But their contribution had both theoretical and practical importance.
It assisted in solving many economic problems, to price derivatives and
to valuation in other areas. Their method has been used to determine
the value of currency options, interest rate options, options on
futures, and so on.
Today, we no longer use the original formula. The interest rate in
modern theories is stochastic, the volatility of the share price varies
stochastically over time, prices develop in jumps, transaction costs
are taken into account and prices can be controlled (e.g. currencies
are restricted to move inside bands in many countries).
Specific Applications of the Formula: Corporate Liabilities
A share can be thought of as an option on the firm. If the value of the
firm is lower than the value of its maturing debt, the shareholders
have the right, but not the obligation, to repay the loans. We can,
therefore, use the Black and Scholes to value shares, even when are not
traded. Shares are liabilities of the firm and all other liabilities
can be treated the same way.
In financial contract theory the methodology has been used to design
optimal financial contracts, taking into account various aspects of
bankruptcy law.
Investment evaluation Flexibility is a key factor in a successful
choice between investments. Let us take a surprising example: equipment
differs in its flexibility - some equipment can be deactivated and
reactivated at will (as the market price of the product fluctuates),
uses different sources of energy with varying relative prices (example:
the relative prices of oil versus electricity), etc. This kind of
equipment is really an option: to operate or to shut down, to use oil
or electricity).
The Black and Scholes formula could help make the right decision.
Guarantees and Insurance Contracts
Insurance policies and financial (and non financial) guarantees can be
evaluated using option-pricing theory. Insurance against the
non-payment of a debt security is equivalent to a put option on the
debt security with a strike price that is equal to the nominal value of
the security. A real put option would provide its holder with the right
to sell the debt security if its value declines below the strike price.
Put differently, the put option owner has the possibility to limit his losses.
Option contracts are, indeed, a kind of insurance contracts and the two markets are competing.
Complete Markets
Merton (1977) extend the dynamic theory of financial markets. In the
1950s, Kenneth Arrow and Gerard Debreu (both Nobel Prize winners)
demonstrated that individuals, households and firms can abolish their
risk: if there exist as many independent securities as there are future
states of the world (a quite large number). Merton proved that far
fewer financial instruments are sufficient to eliminate risk, even when
the number of future states is very large.
Practical Importance
Option contracts began to be traded on the Chicago Board Options
Exchange (CBOE) in April 1973, one month before the formula was
published.
It was only in 1975 that traders had begun applying it - using
programmed calculators. Thousands of traders and investors use the
formula daily in markets throughout the world. In many countries, it is
mandatory by law to use the formula to price stock warrants and
options. In Israel, the formula must be included and explained in every
public offering prospectus.
Today, we cannot conceive of the financial world without the formula.
Investment portfolio managers use put options to hedge against a
decline in share prices. Companies use derivative instruments to fight
currency, interest rates and other financial risks. Banks and other
financial institutions use it to price (even to characterize) new
products, offer customized financial solutions and instruments to their
clients and to minimize their own risks.
Some Other Scientific Contributions
The work of Merton and Scholes was not confined to inventing the formula.
Merton analysed individual consumption and investment decisions in
continuous time. He generalized an important asset pricing model called
the CAPM and gave it a dynamic form. He applied option pricing formulas
in different fields.
He is most known for deriving a formula which allows stock price movements to be discontinuous.
Scholes studied the effect of dividends on share prices and estimated
the risks associated with the share which are not specific to it. He is
a great guru of the efficient marketplace ("The Invisible Hand of the
Market").
