Turning Finance into Science:
Risk Management and the Black-
Scholes Options Pricing Model
Albert Kim
Mary Frauley
Writing for the Sciences
English ENG-LBE 09
Monday, May 1
st
2000
Recently, the people behind the famed Black- Scholes Options
Pricing Model received the Nobel Prize in Economics. Scientific
American delves into this formula that has its share of praise,
and criticism
A New Breed of $cience
In recent years, a new discipline called financial engineering has emerged
in order to attempt to understand finance using a scientific approach.
Mathematicians, physicists and traders work together in this discipline in order
to incorporate the use of advanced mathematics with everyday finance (Stix,
1998).
Although financial engineering deals with many aspects of finance, the
main application of this discipline is risk management within the stock market.
Regardless of what type of stock market transaction one performs, risk is
always present. However, it is the management of this risk that is studied by
these “financial engineers”. People need a fast and reliable way to calculate and
control the risk involved in all their stock trading.
This is where the Black- Scholes Option Pricing Model comes in. This
ideas behind this formula, created by Prof. Robert C. Merton, Prof. Myron S.
Scholes and the late Fisher Black, has been described by one economist as “the
most successful theory not only in finance but in all of economics.” (Stix, 1998)
Options

financial world. This is what the Black- Scholes Options Pricing Model does.
The Math Behind It
Option pricing requires five inputs: the option’s exercise price, the time
to expiration, the price of the stock at the time of evaluation, current interest
rates and the volatility of the stock (Dammers, 1998). The only unreliable
factor is the volatility of the stock. This number can be estimated from
market data (Stix, 1998). The formula is as follows:
where the variable d is defined by:
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According to this formula, the value of the 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
higher the current share price S, the higher the volatility of the share price
(Greek letter) sigma , the higher the interest rate r, the longer the time until the
call date t and the lower the strike price L, the higher the value of the option C
will be.
Limitations of the Model
As consistent as the model, there are limitations to the model. One
limitation is that it assumes that the options can only be exercised on the call
date. In other words, it cannot be exercised earlier. This model involves
“European- Style” options, rather than “American- Style” options. “American-
Style” options can be exercised anytime (Dammers, 1998). American options
are more flexible, thus more valuable. The model only takes European- style
options into account. Thus, the model underestimates the value of options.
Most options are not exercised until the call date anyways, but this law is not
written in stone.
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Another limitation is that it assumes that the interest rate (determined
by the U.S. Government) is known and will remain more- or- less constant.
Many researchers have concluded that this is a safe assumption to make. But

excessive risk by means of pricing options. However, this formula contained
financially unrealistic assumptions, such as the existence of negative values
for stock prices and a zero interest rate (Stix, 1998). His paper was shelved
and went unnoticed for decades.
It wasn’t until 1955 that the idea of options pricing resurfaced, when a
professor at the Massachusetts Institute of Technology named Paul Samuelson
browsed through the Sorbonne library. He began developing a formula of his
own. Other mathematicians, such as Case Sprenkle and James Boness began
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toying with Bachelier’s ideas as well (Royal Swedish Academy of Sciences,
1997). But all of their efforts went fruitless.
A Revolution
Then in 1968, a 31- year- old independent finance contractor named
Fisher Black and a 28- year- old assistant professor of finance at MIT named
Myron Scholes (Rubash, 1998) began their work on options pricing. They were
dissatisfied with all the formulas that had preceded them, because they were
overly complicated and made assumptions that didn’t make sense. They
wanted to find a formula that would calculate the fair price of an option at any
moment in time just by knowing the current price of the stock, but they
couldn’t see their way through the mass of equations they had inherited
(NOVA Online, 2000).
Then they decided to try something different. They decided to strip
previously derived formulas to their bare- boned state. They dropped
everything that represented something un- measurable (NOVA Online, 2000).
They were left with the vitals of calculating an option: the option’s exercise
price, the time to expiration, the price of the stock at the time of evaluation,
current interest rates and the volatility of the stock. But they were stuck with
one problem: one couldn’t measure volatility, or in other words, risk.
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So they decided if they couldn’t measure the risk of an option, they

changes over time, the investor must alter the composition of the portfolio,
the ratio of number of shares to the number of options, to ensure that the
holdings remain without risk (Stix, 1998).
They made up a theoretical portfolio of stocks and options. Whenever
either fluctuated up or down, they tried to hedge against the movement by
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making another move in the opposite direction. Their aim was to keep the
overall value of the portfolio in perfect balance. In other words, they tried to
minimize risk.
They discovered that they could indeed reduce risk by creating a balance
in which all movements in the markets cancelled each other out. Black and
Scholes had found a theoretical way to neutralize risk (NOVA Online, 2000).
With risk now virtually eliminated from their equation, they had a
mathematical formula that could give them the price of any option. This was a
marvelous achievement.
There was a practical problem with their formula. It assumed that
markets were always in equilibrium, that supply equals demand (NOVA Online,
2000). They needed a way to instantly rebalance a portfolio of stocks and
options to keep countering all their movements.
A Harvard graduate by the name of Robert Merton solved this problem
by introducing the notion of continuous time. This idea is rooted in rocket
science. A Japanese mathematician by the name of Kiyosi Ito theorized that
when you plot the trajectory of a rocket, knowing where the rocket was
second- by- second was not enough. You needed to know where the rocket
was continuously. So he broke time down into infinitely small increments,
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thus smoothening the graphing of its path out until it became a continuum so
that the trajectory could be constantly updated (NOVA Online, 2000). Merton
applied this idea to the Black- Scholes model so that the value of an option
could be constantly recalculated and risk eliminated continually (NOVA Online,

match traders’ skill and gut intuition about market dynamics (Royal Swedish
Academy of Sciences, 1997).
Derivatives make the news because, like an airplane crash, their losses
can dramatic and chaotic. Enormous losses by Proctor & Gamble and Gibson
Greetings and the bankruptcies of Barings Bank and Orange County, California
have been attributed to the use of models (Stix, 1998).
Fig 3: Derivatives Debacles [Stix, G. (1998, May). A Calculus of Risk. Scientific American ,
pp. 91.]
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However, Scholes says that it was not so much the formula itself that
caused these losses, rather its misuse by market traders. Every statistician
and mathematician knows you cannot predict the future with 100% accuracy.
Laws as rigid as the laws of physics do not govern the market. Peter Fisher, a
New York economist says: “Math doesn’t drive financial markets, people drive
financial markets, and people are not predictable. We do not yet have a
universal theory of human behavior or human motivation.” (NOVA Online,
2000)
It was not the model by itself that caused these losses, but the blind
faith that market traders put into it. They all jumped at the prospect of
making money without risk. However, this formula cannot eliminate risk, it
can only minimize it. Like many mathematical models, it relies on imputs and
assumes a functioning market. It is a powerful way to manage risk, but it’s not
a crystal ball. Scholes says this equation should be used as a tool for making
decisions, not a platform from which all decisions should be made (NOVA
Online, 2000).
Fisher says: “If a random bolt of lightning hits you when you’re
standing in the middle of the field, that feels like a random event. But if your
business is to stand in random fields during lightning storms, then you should
anticipate, perhaps a little more robustly, the risks you’re taking on.” (NOVA
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NOVA Online. (Feb 8 2000) “Trillion Dollar Bet”. Public Broadcasting
Syndicate Homepage. [Online] 4.9 (2000): Available URL:
/>Ross, Sheldon M. (1999). An Introduction to Mathematical Finance: Options
and other Topics. Cambridge, MA: Cambridge University Press.
Royal Swedish Academy of Sciences. (1997) “Additional background
material on the Bank of Sweden Prize in Economic Sciences in Memory of Alfred
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Nobel 1997.” The Official Website of the Nobel Foundation [Online] 4.9 (2000):
Available URL: 97/ecoback97.html
Rubash, Kevin. (1998) “A Study of Option Pricing Models” Bradley
University. [Online] 4.9 (2000): Available URL: />7Earr/bsm /m odel.html
Stix, G. (1998, May). A Calculus of Risk. Scientific American , pp. 90- 98.
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