By Will Kenton for Investopedia
What
Is the Black Scholes Model?
The Black Scholes model, also known as
the Black-Scholes-Merton (BSM) model, is a mathematical model for pricing an
options contract. In particular, the model estimates the variation over time of
financial instruments. It assumes these instruments (such as stocks or futures)
will have a lognormal distribution of prices. Using this assumption and
factoring in other important variables, the equation derives the price of a
call option.
KEY
TAKEAWAYS
- The
Black-Scholes Merton (BSM) model is a differential equation used to solve
for options prices.
- The
model won the Nobel prize in economics.
- The
standard BSM model is only used to price European options and does not
take into account that U.S. options could be exercised before the
expiration date.
The Basics of the Black Scholes Model
The model assumes the price of heavily traded assets follows 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. Also called Black-Scholes-Merton, it was the first widely used model for option pricing. It's used to calculate the theoretical value of options using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration and expected volatility.
The formula, developed by three economists—Fischer Black, Myron Scholes and
Robert Merton—is perhaps the world's most well-known options pricing
model. The initial equation was introduced in Black and Scholes' 1973
paper, "The Pricing of Options and Corporate Liabilities," published
in the Journal of Political Economy. Black passed away two years before Scholes and Merton
were awarded the 1997 Nobel Prize in economics for their work in finding a new
method to determine the value of derivatives (the Nobel Prize is not given
posthumously; however, the Nobel committee acknowledged Black's role in the
Black-Scholes model).
The Black-Scholes model makes certain
assumptions:
- The option is European and can only be exercised at expiration.
- No
dividends are paid out during the life of the option.
- Markets
are efficient (i.e., market movements cannot be predicted).
- There
are no transaction costs in buying the option.
- The
risk-free rate and volatility of the underlying are known and constant.
- The
returns on the underlying asset are normally distributed.
While the original Black-Scholes model didn't consider the effects of
dividends paid during the life of the option, the model is frequently adapted
to account for dividends by determining the ex-dividend date value of the
underlying stock.
The Black Scholes Formula
The
mathematics involved in the formula are complicated and can be intimidating.
Fortunately, you don't need to know or even understand the math to
use Black-Scholes modeling in your own strategies. Options traders have
access to a variety of online options calculators, and many of today's trading
platforms boast robust options analysis tools, including indicators and
spreadsheets that perform the calculations and output the options pricing
values.
The
Black Scholes call option formula is calculated by multiplying the stock price
by the cumulative standard normal probability distribution function.
Thereafter, the net present value (NPV) of the strike price multiplied by the
cumulative standard normal distribution is subtracted from the resulting value
of the previous calculation.
What
Does the Black Scholes Model Tell You?
The Black Scholes model is one of the
most important concepts in modern financial theory. It was developed in 1973 by
Fischer Black, Robert Merton, and Myron Scholes and is still widely used today. It is
regarded as one of the best ways of determining fair prices of options. The
Black Scholes model requires five input variables: the strike price of an
option, the current stock price, the time to expiration, the risk-free rate,
and the volatility.
The model assumes stock prices follow a lognormal
distribution because asset prices cannot be negative (they are bounded by
zero). This is also known as a Gaussian distribution. Often, asset prices are
observed to have significant right skewness and some degree
of kurtosis (fat tails). This means high-risk downward moves often happen
more often in the market than a normal distribution predicts.
The assumption of lognormal underlying asset prices should thus show that implied volatilities are similar for each strike price according to the Black-Scholes model. However, since the market crash of 1987, implied volatilities for at the money options have been lower than those further out of the money or far in the money. The reason for this phenomena is the market is pricing in a greater likelihood of a high volatility move to the downside in the markets.
This has led to the presence of the volatility skew. When the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen. Thus, the Black-Scholes model is not efficient for calculating implied volatility.Limitations of the Black Scholes Model
As stated previously, the Black Scholes
model is only used to price European options and does not take into account
that U.S. options could be exercised before the expiration date. Moreover, the
model assumes dividends and risk-free rates are constant, but this may not be
true in reality. The model also assumes volatility remains constant over the
option's life, which is not the case because volatility fluctuates with the
level of supply and demand.
Moreover, the model assumes that there are no transaction costs or
taxes; that the risk-free interest rate is constant for all maturities; that
short selling of securities with use of proceeds is permitted; and that there
are no risk-less arbitrage opportunities. These assumptions can lead to prices
that deviate from the real world where these factors are present.
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