Black-Scholes Model


Black-Scholes Model - Definition


A mathematical formula designed to price an option as a function of certain variables-generally stock price, striking price, volatility, time to expiration, dividends to be paid, and the current risk-free interest rate.

Black-Scholes Model - Introduction


The Black-Scholes model is a tool for equity options pricing. Prior to the development of the Black-Scholes Model, there was no standard options pricing method and nobody can put a fair price to charge for options. The Black-Scholes Model turned that guessing game into a mathematical science which helped develop the options market into the lucrative industry it is today. Options traders compare the prevailing option price in the exchange against the theoretical value derived by the Black-Scholes Model in order to determine if a particular option contract is over or under valued, hence assisting them in their options trading decision. The Black-Scholes Model was originally created for the pricing and hedging of European Call and Put options as the American Options market, the CBOE, started only 1 month before the creation of the Black-Scholes Model. The difference in the pricing of European options and American options is that options pricing of European options do not take into consideration the possibility of early exercising. American options therefore command a higher price than European options due to the flexibility to exercise the option at anytime. The classic Black-Scholes Model does not take this extra value into consideration in its calculations.


Black-Scholes Model Assumptions


There are several assumptions underlying the Black-Scholes model of calculating options pricing. The most significant is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. The Black-Scholes model also assumes stocks move in a manner referred to as a random walk; at any given moment, they are as likely to move up as they are to move down. These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer.

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

As you can see, the validity of many of these assumptions used by the Black-Scholes Model are questionable or invalid, resulting in theoretical values which are not always accurate. Hence, 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 under priced nature of a stock option.

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Black-Scholes Model Inputs


The Black-Scholes model takes as input current prices, the option's strike price, length of time until the option expires worthless, an estimate of future volatility known as implied volatility, and risk free rate of return, generally defined as the interest rate of short term US treasury notes. The Black-Scholes Model also works in reverse: instead of calculating a price, an implied volatility for a given price can be calculated. Implied Volatility is commonly calculated using the Black-Scholes Model in order to plot the Volatility Smile or Volatility Skew.

The mathematical characteristics of the Black-Scholes model are named after 5 greek letters used to represent them in equations; Delta, Gamma, Vega, Theta and Rho. These are now passionately known to option traders as the "Greeks".


Known Problems Of The Black-Scholes Model


First, the Black-Scholes Model assumes that the risk-free rate and the stock's volatility are constant. This is obviously wrong as risk free rate and volatility fluctuates according to market conditions.

Second, 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.

Third, the Black-Scholes Model assumes a stock pays no dividends until after expiration.

Fourth, analysts can only estimate a stock's volatility instead of directly observing it, as they can for the other inputs.

Fifth, the Black-Scholes Model tends to overvalue far out-of-the-money calls and undervalue deep in-the-money calls.

Sixth, the Black-Scholes Model tends to misprice options that involve high-dividend stocks.

To deal with these limitations, a Black-Scholes Model variant known as ARCH, Autoregressive Conditional Heteroskedasticity, was developed. This variant replaces constant volatility with stochastic (random) volatility. A number of different models was developed afterwhich like the GARCH, E-GARCH, N-GARCH, H-GARCH, etc, 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.


Alternative to the 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. Let's remember that the Black-Scholes model was initially invented for the purpose of pricing European style options. As such, a new options pricing model called the Cox-Rubinstein binomial model. It is commonly known as the Binomial Option Pricing Model or simply, the Binomial Model was invented in 1979. This options pricing model was more appropriate for American Style options as it allows for the possibility of early exercise.


OppiE's Note For the amateur, beginner option trader, it suffices to know that you can determine if a stock option you are about to buy is over-valued by comparing it with the theoretical value derived by the Black-Scholes Model. As the Black-Scholes model typically undervalues stock options, an option should not be deemed over-valued unless it is grossly higher than the theoretical value produced by the Black-Scholes model. The prevailing price of a stock option is usually much higher than its theoretical value due to implied volatility. Buying such an option could subject an option position to losses should implied volatility drop in the near future.



The Black-Scholes Formula Is:



C0 = S0N(d1) - Xe-rTN(d2)
 
Where:
d1 = [ln(S0/X) + (r + σ2/2)T]/ σ √T
 
And:
d2 = d1 - σ √T

And where:
C0 = current option value
S0 = current stock price
N(d) = the probability that a random draw from a standard normal distribution will be less than (d).
X = exercise price
e = 2.71828, the base of the natural log function
r = risk-free interest rate (annualized continuously compounded rate on a safe asset with the same maturity as the expiration of the option; usually the money market rate for a maturity equal to the option's maturity.)
T = time to option's maturity, in years
ln = natural logarithm function
σ = standard deviation of the annualized continuously compounded rate of return on the stock


Even though the original Black-Scholes Model do not take dividends into consideration, an extension of the Black-Scholes Model proposed by Merton in 1973 alters the Black-Scholes model in order to take annual dividend yield into consideration. This model is not as widely used as the original Black-Scholes Model as not every company pays dividends.

Here's the Black-Scholes Model for Dividend Stocks formula :

C0 = Se-dTN(d1) - Xe-rTN(d2)


Where:
d1 = [ln(S0/X) + (r - d + σ2/2)T]/ σ √T


And:
d2 = d1 - σ √T



Black-Scholes Model Calculator



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History Of The Black-Scholes model

The Black-Scholes Model was first discovered in 1973 by Fischer Black and Myron Scholes, and then further developed by Robert Merton. It was for the development of the Black-Scholes Model that Scholes and Merton received the Nobel Prize of Economics in 1997 (Black had passed away two years earlier). 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.

Black-Scholes Model creators, Myron Scholes was then the professor of finance in Stanford University, Robert Merton was an economist with Harvard University and Fischer Black was a mathematical physicist with a doctorate from Harvard. When Myron Scholes and Fischer Black tried to publish their idea of the Black-Scholes model in 1970, Chicago University's Journal of Political Economy and Harvard's Review of Economics and Statistics both rejected the paper. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the Black-Scholes Model paper. Within six months of the publication of the Black-Scholes Model article, Texas Instruments had incorporated the Black-Scholes Model into their calculator, announcing the new feature with a half-page ad in The Wall Street Journal.

The three young Black-Scholes Model researchers -- who were then in their twenties -- set about trying to find an answer to derivatives pricing using mathematics, exactly the way a physicist or an engineer approaches a problem. However, few think their approach will work because options trading was very new then and was highly volatile (The CBOE opened in April 1973, just one month prior to the release of the Black-Scholes Model paper!). The skeptics were all wrong. Mathematics could be applied using a little known technique known as stochastic differential equations and that discovery led to the development of the Black-Scholes Model that we know today.


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Creators Of The Black-Scholes model

Myron Scholes, co-developer of the Black-Scholes Model Myron Scholes was born in Timmins, Ontario, Canada on July 1, 1941. After attending McMaster University in Hamilton, Ontario (B.A., 1961), Scholes studied under Nobel laureate Merton H. Miller at the University of Chicago (M.B.A., 1964; Ph.D., 1970). Scholes taught at the Massachusetts Institute of Technology (196873) and the University of Chicago (197383) before joining Stanford University in 1983 as a professor of both law and finance, becoming emeritus in 1996. He also worked with many economic and financial institutions, including the National Bureau of Economic Research, Salomon Brothers, and Long-Term Capital Management (LTCM), which Merton cofounded in 1994.


Robert Merton, co-developer of the Black-Scholes Model Robert Merton was born in New York, on July 31, 1944. After studying engineering mathematics at Columbia University (B.S., 1966) and applied mathematics at the California Institute of Technology (M.S., 1967), Merton turned to the study of economics at the Massachusetts Institute of Technology (Ph.D., 1970). He taught at MIT's Sloan School of Management from 1970 until 1988, when he 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. Merton wrote many economic treatises, as well as the book Continuous-Time Finance (1990).


Fischer Black, co-developer of the Black-Scholes Model Fischer Black was born on January 11, 1938. He received a Ph.D. in Applied Math from Harvard University in 1964. He was a student of Marvin Minsky and worked on problems in Artificial Intelligence. In 1971 he began to work at the University of Chicago. He later left the University of Chicago to work at the MIT Sloan School of Management. In 1984 he joined Goldman Sachs.



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Binomial Option Pricing Model

Binomial Option Pricing Model (BOPM) was invented by Cox-Rubinstein in 1979. It 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. The binomial model is thus named as it returns 2 possibilities at any given time. Therefore, instead of assuming that an option trader will hold an option contract all the way to expiration like in the Black-Scholes Model, it calculates the value of that trader exercising that option contract with every possible future up and down moves on its underlying asset, reflecting its effects on the present value of that option, thus giving a more accurate theoretical price of an American Style option.

The binomial model produces a binomial distribution of all the the possible paths that a stock price could take during the life of the option. A binomial distribution, or simply known as a "Binomial Tree", assumes that a stock can only increase or decrease in price all the way until the option expires and then maps it out in a "tree". Here is a simplied version of a binomial distribution just for illustration purpose :
simplified binomial distribution
It then fills in the theoretical value of that stock's options at each time step from the very bottom of the binomial tree all the way to the top where the final, present, theoretical value of a stock option is arrived. Any adjustments to stock prices at an ex-dividend date or option prices as a result of early exercise of American options are worked into the calculations at each specific time step .

Advantage Of The Binomial Option Pricing Model
It can more accurately price American Style Options than the Black-Scholes Model as it takes into consideration the possibilities of early exercise and other factors like dividends.

Disadvantage Of The Binomial Option Pricing Model
As it is much more complex than the Blac-Scholes Model, it is slow and not useful for calculating thousands of option prices quickly.

Binomial Option Pricing Model Calculator
An excellent Binomial Model Calculator can be found here.

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