What are the assumptions behind the Black-Scholes model?

The Black-Scholes Option Pricing Model is an approach used for calculating the value of a stock option. It can be used to calculate values of both call and put options. The Black-Scholes model is described in detail at this page: Black-Scholes model.

This page provides an overview of assumptions underlying the Black-Scholes model.

Knowing the assumptions to the Black-Scholes model is important for its correct application. Many of the assumptions mentioned below are invalidated in the real world; therefore, just blindly applying the Black-Scholes formula to real-world situations can possibly lead to incorrect numbers.

What are the assumptions behind the Black-Scholes model?

There are several assumptions underlying the Black-Scholes model.

1) Constant volatility. The most significant assumption is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. While volatility can be relatively constant in very short term, it is never constant in longer term. Some advanced option valuation models substitute Black-Schole's constant volatility with stochastic-process generated estimates.

2) Efficient markets. This assumption of the Black-Scholes model suggests that people cannot consistently predict the direction of the market or an individual stock. The Black-Scholes model assumes stocks move in a manner referred to as a random walk. Random walk means that at any given moment in time, the price of the underlying stock can go up or down with the same probability. The price of a stock in time t+1 is independent from the price in time t.

3) No dividends. Another assumption is that the underlying stock does not pay dividends during the option's life. In the real world, most companies pay dividends to their share holders. The basic Black-Scholes model was later adjusted for dividends, so there is a workaround for this. This assumption relates to the basic Black-Scholes formula. A common way of adjusting the Black-Scholes model for dividends is to subtract the discounted value of a future dividend from the stock price.

4) Interest rates constant and known. The same like with the volatility, interest rates are also assumed to be constant in the Black-Scholes model. The Black-Scholes model uses the risk-free rate to represent this constant and known rate. In the real world, there is no such thing as a risk-free rate, but it is possible to use the U.S. Government Treasury Bills 30-day rate since the U. S. government is deemed to be credible enough. However, these treasury rates can change in times of increased volatility.

5) Lognormally distributed returns. The Black-Scholes model assumes that returns on the underlying stock are normally distributed. This assumption is reasonable in the real world.

6) European-style options. The Black-Scholes model assumes European-style options which can only be exercised on the expiration date. American-style options can be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility.

8) Liquidity. The Black-Scholes model assumes that markets are perfectly liquid and it is possible to purchase or sell any amount of stock or options or their fractions at any given time.

See the Black-Scholes model page for more details about the Black-Scholes model and to read about how these assumptions relate to real-world scenarios.

Is there anything else I should know about?

The next page called Black-Scholes formula option value on-line calculator provides as the title suggest an online calculator for the Black-Scholes formula.

The so-called put-call parity is another topic directly related to Black-Scholes.