Using Financial Options and Derivatives
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
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
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
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
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
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
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
Put differently, the put option owner has the possibility to limit
Option contracts are, indeed, a kind of insurance contracts and the
two markets are competing.
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.
Option contracts began to be traded on the Chicago Board Options
Exchange (CBOE) in April 1973, one month before the formula was
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
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").
AUTHOR BIO (must be included with the article)
Sam Vaknin ( samvak.tripod.com ) is the author of Malignant
Self Love - Narcissism Revisited and After the Rain - How the West
Lost the East. He served as a columnist for Global Politician,
Central Europe Review, PopMatters, Bellaonline, and eBookWeb, a
United Press International (UPI) Senior Business Correspondent, and
the editor of mental health and Central East Europe categories in
The Open Directory and Suite101.
Until recently, he served as the Economic Advisor to the Government
Visit Sam's Web site at http://samvak.tripod.com