Black Scholes Model

**History**

The Black-Scholes Model, also known as the *Black-Scholes-Merton Model, * was first discovered in 1973 by Fischer Black and Myron Scholes, and then further developed by Robert Merton.

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black started out working to create a valuation model for stock warrants. Soon after this discovery, Myron Scholes joined Black and the result of their work is a pricing model we use today which is surprisingly accurate.

Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model comes in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.

The idea of the Black-Scholes Model was first published in "The Pricing of Options and Corporate Liabilities" of the Journal of Political Economy by Fischer Black and Myron Scholes and then elaborated in "Theory of Rational Option Pricing" by Robert Merton in 1973.

**Fischer Black**

Born: 1938 Died: August 30, 1995

1959 -- Earned bachelor's degree in physics

1964 -- Earned PhD. from Harvard in applied math

1971 -- Joined University of Chicago Graduate School of Business

1973 -- Published "The Pricing of Options and Corporate Liabilities"

19?? -- Left the University of Chicago to teach at MIT

1984 -- Left MIT to work for Goldman Sachs & Co.

**Myron Scholes**

Born: 1941

1962 - Bachelor's degree in Economics from McMaster University

1964 - MBA from the University of Chicago

1969 -Ph.D. from the University of Chicago

1973 -- Published "The Pricing of Options and Corporate Liabilities". Also moved to the University of Chicago Graduate School of Business.

1981 – Teaching at Stanford University.

1990 - Works in the derivatives trading group at Salomon Brothers.

1996 – Retired from teaching

1997 - Shared the Nobel Prize in Economics with Robert C. Merton "for a new method to determine the value of derivatives".
Scholes is currently the chairman of Platinum Grove Asset Management, a hedge fund, which he started with former LTCM partner Chi-fu Huang.

**Robert Merton**

Born: July 31, 1944

1966 – B.S. - Columbia University

1967 – M.S. - California Institute

1970 - Studied economics at the Massachusetts Institute of Technology

1970 – 1988 - Taught at MIT's Sloan School of Management

1988 - Joined the faculty of the Harvard Business School. In addition to his academic duties, he served on the editorial boards of numerous economic journals and as a principal member of Long-Term Capital Management, an investment firm he cofounded and in which Scholes was also a partner.

1990 – Published “Continuous-Time Finance”

Merton also wrote many other economic treatises.

The Black Scholes Model is one of the most important concepts in modern financial theory. The Black Scholes Model is considered the standard model for valuing options. A model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry.
Fortunately one does not have to know calculus to use the Black Scholes model.

**Black-Scholes Model Assumptions**

There are several assumptions underlying the Black-Scholes model of calculating options pricing..

The exact 6 assumptions of the Black-Scholes Model are :

1. Stock pays no dividends.

2. Option can only be exercised upon expiration.

3. Market direction cannot be predicted, hence "Random Walk."

4. No commissions are charged in the transaction.

5. Interest rates remain constant.

6. Stock returns are normally distributed, thus volatility is constant over time.

These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer.

As you can see, many assumptions of the Black-Scholes Model are invalid, resulting in theoretical values which are not always accurate. Therefore, theoretical values derived from the Black-Scholes Model are only good as a guide for relative comparison and is not an exact indication to the over- or underpriced nature of a stock option.

**Limitations of the Black Scholes Model**

The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely used as a useful approximation, but proper use requires understanding its limitations – blindly following the model exposes the user to unexpected risk.

Among the most significant limitations are:

1. The Black-Scholes Model assumes that the risk-free rate and the stock’s volatility are constant.

2. The Black-Scholes Model assumes that stock prices are continuous and that large changes (such as those seen after a merger announcement) don’t occur.

3. The Black-Scholes Model assumes a stock pays no dividends until after expiration.

4. Analysts can only estimate a stock’s volatility instead of directly observing it, as they can for the other inputs.

5. The Black-Scholes Model tends to overvalue deep out-of-the-money calls and undervalue deep in-the-money calls.

6. The Black-Scholes Model tends to misprice options that involve high-dividend stocks.

To deal with these limitations, a Black-Scholes variant known as ARCH, Autoregressive Conditional Heteroskedasticity, was developed. This variant replaces constant volatility with stochastic (random) volatility. A number of different models have been developed all incorporating ever more complex models of volatility. However, despite these known limitations, the classic Black-Scholes model is still the most popular with options traders today due to its simplicity.

**Variants of the Black Scholes Model**

There are a number of variants of the original Black-Scholes model. As the Black-Scholes Model does not take into consideration dividend payments as well as the possibilities of early exercising, it frequently under-values Amercian style options.

As the Black-Scholes model was initially invented for the purpose of pricing European style options a new options pricing model called the Cox-Rubinstein binomial model is also used. It is commonly known as the *Binomial Option Pricing Model* or more simply, the Binomial Model, which was invented in 1979. This options pricing model was more appropriate for American Style options as it allows for the possibility of early exercise.

*The Binomial Option Pricing Model (BOPM)*, invented by Cox-Rubinstein, was originally invented as a tool to explain the Black-Scholes Model to Cox's students. However, it soon became apparent that the binomial model is a more accurate pricing model for American Style Options.

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