Black-Scholes is a way of modeling stock prices. The Black-Scholes model focuses on the current stock price and the volatility of the stock price to determine the likelihood of future price moves. This type of analysis is most often used to accurately price stock options based on the price of the underlying stock. The model is used to find the current value of a call option whose ultimate value depends on the price of the stock at the expiration date. Because the stock price keeps changing, the value of this call option will change too.

This type of analysis differs from traditional technical analysis because it does not select a direction for a move. Linear regression and Gann angles, for example, assume that a stock price will move in a straight line. A longer, straighter line means a stronger trend, so we can confidently predict that the stock price will continue to rise or fall at the same rate. Black-Scholes says that a stock chart with a price rising or falling in a straight line is very unlikely. A longer, straighter line is even less likely.

Bollinger Bands come closer to the Black-Scholes model. The standard deviation formula used in Bollinger Bands is similar to the volatility formula used by the Black-Scholes model. For that reason, neither one expects a sharp rise or drop in price to continue forever. If you see a stock chart with a pattern like that, and you add Bollinger Bands to that stock chart, then you will see the bands get wider and wider over time. The Bollinger Bands, like the Black-Scholes model, are saying that a stock like that becomes harder and harder to predict.

Black-Scholes differs, however, from Bollinger Bands, because Bollinger Bands expect an extreme move to be followed by a pullback. The center of the Bollinger Bands is set by a moving average, under the assumption that the price is likely to return toward that mean. Black-Scholes always centers its predictions around the current price, as in the efficient market theorem.

How does the Black-Scholes formula work?

The Black-Scholes formula expresses the value of a call option by taking the current stock prices multiplied by a probability factor (D1) and subtracting the discounted exercise payment times a second probability factor (D2).

Black-Scholes posits that instruments, such as stock shares or futures contracts, will have a lognormal distribution of prices following a random walk with constant drift and volatility. Using this assumption and factoring in other important variables, the equation derives the price of a European-style call option.

The Black-Scholes equation requires five variables. These inputs are volatility, the price of the underlying asset, the strike price of the option, the time until expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for options sellers to set rational prices for the options that they are selling.

Furthermore, the model predicts that 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.

Black-Scholes Assumptions
The Black-Scholes model makes certain assumptions:

What Are the Limitations of the Black-Scholes Model?
The Black-Scholes model is only used to price European options and does not take into account that American options could be exercised before the expiration date. Moreover, the model assumes dividends, volatility, and risk-free rates remain constant over the option’s life.

• No dividends are paid out during the life of the option.
• Markets are random (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 asset are known and constant.
• The returns of the underlying asset are normally distributed.
• The option is European and can only be exercised at expiration.
• 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 model is also modified by many option-selling market makers to account for the effect of options that can be exercised before expiration.

Not taking into account taxes, commissions or trading costs or taxes can also lead to valuations that deviate from real-world results.