The Black-Scholes model is one of the most important concepts in modern financial theory. Also known as the Black-Scholes-Merton (BSM) model, this mathematical equation estimates the theoretical value of derivatives based on other investment instruments, taking the impact of time and other risk factors into account. It was developed in 1973 and is still regarded as one of the best ways to price an options contract .

Key Takeaways

  • The Black-Scholes model is also known as the Black-Scholes-Merton or BSM model.
  • It's a differential equation that's widely used to price options contracts.
  • 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 Black-Scholes model is usually accurate but it makes certain assumptions that can lead to predictions that deviate from real-world results.
  • The standard BSM model is only used to price European options because it doesn't take into account that American options could be exercised before the expiration date.

History of the Black-Scholes Model

Developed in 1973 by Fischer Black, Robert Merton , and Myron Scholes , the Black-Scholes model was the first widely used mathematical method to calculate the theoretical value of an option contract. It uses current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration, and expected volatility.

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 . Robert C. Merton helped edit the paper. He published his own article in The Bell Journal of Economics and Management Science later that year: "Theory of Rational Option Pricing." He expanded the mathematical understanding and applications of the model and coined the term "Black–Scholes theory of options pricing."

Scholes and Merton were awarded the Nobel Memorial Prize in Economic Sciences in 1997 for their work in finding "a new method to determine the value of derivatives." Black had passed away two years earlier so he could not be a recipient because Nobel Prizes are not given posthumously. The Nobel Committee acknowledged his role in the Black-Scholes model, however.

How the Black-Scholes Model Works

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. The equation uses this assumption and factors in other important variables to derive the price of a European-style call option .

The Black-Scholes equation requires six variables: volatility , the price of the underlying asset, the strike price of the option, the time until the expiration of the option, the risk-free interest rate, and the type of option, whether it's call or put. It's theoretically possible for options sellers to set rational prices with these variables for the options that they're selling.

The model predicts that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. It 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 when it's applied to a stock option.

The Black-Scholes model is often contrasted against the binomial model or a Monte Carlo simulation.