1. Options Pricing: Introduction
  2. Options Pricing: A Review of Basic Terms
  3. Options Pricing: The Basics of Pricing
  4. Options Pricing: Intrinsic Value and Time Value
  5. Options Pricing: Factors That Influence Option Price
  6. Options Pricing: Distinguishing Between Option Premiums and Theoretical Value
  7. Options Pricing: Modeling
  8. Options Pricing: Black-Scholes Model
  9. Options Pricing: Cox-Rubinstein Binomial Option Pricing Model
  10. Options Pricing: Put/Call Parity
  11. Options Pricing: Profit and Loss Diagrams
  12. Options Pricing: The Greeks
  13. Options Pricing: Conclusion

The Black-Scholes formula (also called  Black-Scholes-Merton) was the first widely used model for option pricing. It's used to calculate the theoretical value of European-style options using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration and expected volatility. 

[ Option pricing is very complex because it depends on so many different factors. The good news is that many of these calculations are boiled down into the Greeks (delta, vega, etc.) and each of these Greeks has a specific meaning. If you want to learn more about options trading, check out Investopedia's Options for Beginners Course. You will learn how to interpret expiration dates, distinguish intrinsic value from time value, and much more in over five hours of on-demand video, exercises, and interactive content. ] 

The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model. It was introduced in their 1973 paper, "The Pricing of Options and Corporate Liabilities," published in the Journal of Political Economy. Black passed away two years before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives (the Nobel Prize is not given posthumously; however, the Nobel committee acknowledged Black's role in the Black-Scholes model). 

The Black-Scholes model makes certain assumptions:

  • The option is European and can only be exercised at expiration.
  • No dividends are paid out during the life of the option.
  • Markets are efficient (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 are known and constant.
  • The returns on the underlying are normally distributed.

Note: 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.

Black-Scholes Formula

The formula, shown in Figure 4, takes the following variables into consideration:

  • current underlying price
  • options strike price
  • time until expiration, expressed as a percent of a year
  • implied volatility
  • risk-free interest rates
Figure 4: The Black-Scholes pricing formula for call options.

The model is essentially divided into two parts: the first part, SN(d1), multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright. The second part, N(d2)Ke-rt, provides the current value of paying the exercise price upon expiration (remember, the Black-Scholes model applies to European options that can be exercised only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.

The mathematics involved in the formula are complicated and can be intimidating. Fortunately, you don't need to know or even understand the math to use Black-Scholes modeling in your own strategies. As mentioned previously, options traders have access to a variety of online options calculators, and many of today's trading platforms boast robust options analysis tools, including indicators and spreadsheets that perform the calculations and output the options pricing values. An example of an online Black-Scholes calculator is shown in Figure 5. The user inputs all five variables (strike price, stock price, time (days), volatility and risk free interest rate) and clicks "get quote" to display results. 
 

Black-Scholes option calculator
Figure 5: An online Black-Scholes calculator can be used to get values for both calls and puts. Users  enter the required fields and the calculator does the rest. Calculator courtesy www.tradingtoday.com

 


Options Pricing: Cox-Rubinstein Binomial Option Pricing Model
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