The Black Scholes Model

The Black Scholes pricing model is partially responsible for the options market and options trading becoming so popular. Before it was developed there wasn't a standard method for pricing options, and it was essentially impossible to put a fair value on them. This meant that options weren't commonly viewed as suitable financial instruments by investors and traders, because it was very difficult to determine whether there was good value for money available.

The Black Scholes model changed this; it's a mathematical formula that is designed to calculate a fair value for an option based on certain variables. On this page we provide further information on this model and the role it has to play in options trading. The following topics are covered:

  • History
  • Purpose
  • Input & Assumptions
  • Using the Black Scholes Pricing Model

History

The Black Scholes pricing model is named after the American economists Fischer Black and Myron Scholes. In 1970 Black, a mathematical physicist, and Scholes, a professor of finance at Stanford University, wrote a paper titled “The Pricing of Options and Corporate Liabilities.” They tried to publish the paper, but it was rejected by various publishers, until Chicago University’s Journal of Political Economy agreed to publish it in 1973.

In this paper, Black and Scholes implied that an option had one correct price, which could be determined using an equation that they included in the paper. This equation became known as the Black-Scholes equation or the Black-Scholes formula. Also in 1973, a subsequent paper, “Theory of Rational Option Pricing," was written by Robert Merton, and he expanded on this mathematical approach and introduced the term Black Scholes options pricing model.

At the time, options trading was very new and was considered a very risky and volatile form of trading. Although initially greeted by a great deal of skepticism, Black, Scholes, and Merton showed that mathematics could be applied using differential equations to determine a fair value for European style calls and puts.

The Black Scholes model became widely accepted and it contributed to options trading becoming far more popular than it might otherwise have been. The model is also often referred to as the Black-Scholes-Merton model and is considered to be one of the most significant concepts in modern financial theory. Robert Merton and Myron Scholes were awarded the Nobel Prize in Economics in 1997: two years after the death of Fischer Black.

Purpose

As we have mentioned above, prior to the model it was very difficult for an investor to determine whether or not an option was priced correctly, and therefore whether or not it represented good value. A big part of successful investing and trading is finding opportunities where an asset is underpriced or overpriced and then trading it accordingly. Because this wasn't really possible with options, the market wasn't particularly favored by investors and traders and it was considered very risky.

The Black Scholes formula was developed to calculate an economic value for options that is fair to both the buyer and seller. In theory, if options were bought and sold repeatedly at the price set by this model, then buyers and sellers would both break even on average: not including any commissions charged.

The idea behind the formula is that it's possible to create a perfect hedging situation through combining options contracts and the underlying security, assuming that the contracts are priced correctly.  Basically, the theory proposed that there's only one truly correct price for an option, and that price can be calculated mathematically.

In practice, the price is affected by many factors, including demand and supply, and because of this, options may not always be priced correctly. By using the Black Scholes pricing model, it's possible, theoretically, to determine whether the trading price of an option is higher or lower than it's true value: which can in turn highlight potential trading opportunities.

Inputs & Assumptions

The Black Scholes pricing model is based on a mathematical formula and that formula uses a number of variables or inputs to calculate a fair value for an option. These variables are known as the inputs to the model and they are as follows:

  • The current price of the underlying security
  • The strike price
  • The length of time until expiry
  • The risk free interest rate during the period of the contract
  • The implied volatility of the underlying security

The model also relies on several underlying assumptions for it to work. These assumptions are as follows:

  • The option can only be exercised upon expiration (i.e. it is a European style)
  • The underlying security will sometimes go up in price and sometimes go down and that the direction of the movement cannot be predicted.
  • The underlying security pays no dividends
  • The volatility of the underlying security remains stable during the period of the contract
  • Interest rates remain constant during the period of the contract
  • There are no commissions charged on the purchase or the sale of the option
  • There is no arbitrage opportunity (i.e. neither the buyer nor the seller should gain an immediate benefit)

It should be reasonably obvious that some of these assumptions aren't always going to be valid, and it's very important to recognize this because, it means that there is a distinct possibility that the theoretical values calculated using the Black Scholes model may not be accurate.

Using the Black Scholes Pricing Model

There can be no doubt that the development of the Black Scholes pricing model helped make options trading more viable in the eyes of investors, because it helped to change the idea that valuing options was little more than a guessing game. However, there are a couple of key points you should be aware of.

First, it isn't absolutely necessary to fully understand the mathematical formula behind the pricing model to be successful at options trading and it's not even necessary that you use it at all. If you do wish to use it though, you will probably find it easier to use one of the many Black Scholes model calculating tools on the internet rather than carrying out the calculations yourself. You will find that a number of online brokers include such a calculating tool for their customers to use.

Second, it should be noted that it should never be considered a precise indicator of the true value of an option, because there are some problems with the assumptions that underpin the model. For example, it assumes that interest rates and the volatility of the underlying security will remain constant during the period of the contract, and this is unlikely to be the case.

It also doesn't take into account the fact that some stocks pay dividends, nor the extra value that American style options have because the holder of them is able to exercise them at any point. There are, however, variants of the Black Scholes model that can be applied to factor in such issues.

If you do plan on using the model as part of your trading strategy, then we strongly suggest that you don't rely upon it to return exact values, but rather theoretical values. These theoretical values can then be used for the purposes of comparing options to assist you in determining what trades you should be making. You could also use the model to help decide whether a potential trade you have identified through other methods is likely to be a successful trade or not.

In summary, the Black Scholes pricing model has played a notable part in how the options market and options trading have developed and it certainly still has its uses to traders. You should, however, be fully aware of its limitations and never be entirely dependent on it.