Black-Scholes Option Pricing Model

Black, Scholes, Merton. © KhanAcademy

Option prices are not stagnant and frequently move higher or lower. The casual observer may conclude that such movement is based solely on the changing price of the underlying asset. But that is far from correct.​

There are eight distinct factors that influence the option premium (market price). All but one (future volatility) of those factors is always known to traders. Thus traders can readily calculate the fair value of any option that is based on those factors.

For example

  • Option Type: There are two option types: calls and puts.
  • Strike price is described in the option contract and cannot change.
  • The specific stock (underlying asset).

However, some items that affect the value of an option do change (but these changes are known to everyone):

  • The stock price (underlying asset) changes frequently. 
  • Time: The number of days remaining before the option expires (obviously changes once per day). More sophisticated traders calculate the remaining time in hours or minutes and they do not simply count the number of actual calendar days until the options expire. The novice trader should work with the number of calendar days because it is easily good enough for their purposes.
  • Interest rates. Changes are infrequent and even when they do change, they have a very small effect on most options. Longer-term options are more affected by a change in interest rates.
  • Dividends may be increased or decreased, but that seldom happens more than once per year, and traders can easily change the input to their models.
  • . Volatility is defined as the expected volatility of the underlying asset in the future. See more on this below


Seven of these factors are always known at any specific point in time, and each can be entered into a calculator.

Only one factor is unknown and that is volatility. 

One of the surprising situations that occur when we examine option prices is that they can undergo a very rapid change -- for no apparent reason.

This occurs when the volatility forecast suddenly changes. As an option trader you must be aware of how/why implied volatility can change suddenly.

When option traders use the word 'volatility' it refers to the expected volatility of the underlying stock over a specific period of time. That time slot is from the current moment until the option expires. Because that time represents the future, the volatility cannot be known and must be estimated. Therefore there is one important conclusion that I must emphasize because it is easy to get it wrong.

The option calculator is a valuable tool because it provides good estimates. But, the numbers are still estimates because each value is dependent on the input data. We input a volatility number (that is our estimate for future volatility). Once one number is estimated, all results are only as good as that estimate. NOTE: Individual traders are not expected to to come up with own volatility estimates. Your broker, who provides option values, get them from reasonable sources. But again -- reasonable does not mean the numbers are 100% accurate.

The calculator spits out numbers with several decimal places because that is what computer programmers ask them to do. Do not believe that they the values are that accurate. They are only as good as the volatility estimate.

Conclusion: Use the theoretical option value and each of the Greeks that are provided by the calculator. They are good numbers. But they are estimates. 

Use an option calculator based on the Black-Scholes option pricing model (or on one of the numerous alternatives) to get a solid estimate of  an option's

If you are not familiar with the Greeks, you can get an introduction to the topic here. The Greeks is another topic that frightens many newer traders. Here is the basic truth: The Greeks help you measure risk so that you can manage risk and avoid having unexpected risk for any position. That is all they do. The mathematics may be complicated, but you and I never have to think about that math. These numbers are always provided by brokers or calculators. Using the Greeks is easy. 

Where to find option-pricing calculators (if your broker does  not provide a good one):
NOTE: Choose "American" style options for stocks; Choose "European" style for index options.

  • CBOE (Chicago Board options Exchange)
  • ISE (International Securities Exchange)