Black-Scholes Model

What Is The Black Scholes Model and What Does It Do In Options Trading?

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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.

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.

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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