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-Rubenstein 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 model for calculating the premium of an option was introduced in 1973 in a paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political Economy. The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model. 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 is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option's lifetime. While the original Black-Scholes model did not take into consideration the effects of dividends paid during the life of the option, the model can be adapted to account for dividends by determining the ex-dividend date value of the underlying stock.

The model makes certain assumptions, including:

  • The options are European and can only be exercised at expiration
  • No dividends are paid out during the life of the option
  • Efficient markets (i.e., market movements cannot be predicted)
  • No commissions
  • The risk-free rate and volatility of the underlying are known and constant
  • Follows a lognormal distribution; that is, returns on the underlying are normally distributed.

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 are exercisable 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 is complicated and can be intimidating. Fortunately, however, traders and investors do not need to know or even understand the math to apply Black-Scholes modeling in their 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 must input all five variables (strike price, stock price, time (days), volatility and risk free interest rate).

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

Related Articles
  1. Trading

    Understanding the Black-Scholes Model

    The Black-Scholes model is a mathematical model of a financial market. From it, the Black-Scholes formula was derived. The introduction of the formula in 1973 by three economists led to rapid ...
  2. Trading

    NYIF Instructor Series: Black Scholes Model

    In this short instructional video Anton Theunissen explains the Black Scholes model.
  3. Trading

    Circumvent Limitations of Black-Scholes Model

    Mathematical or quantitative model-based trading continues to gain momentum, despite major failures like the financial crisis of 2008-09, which was attributed to the flawed use of trading models. ...
  4. Trading

    How To Build Valuation Models Like Black-Scholes (BS)?

    Want to build a model like Black-Scholes? Here are the tips and guidelines for developing a framework with the example of the Black-Scholes model.
  5. Trading

    The Anatomy of Options

    Find out how you can use the "Greeks" to guide your options trading strategy and help balance your portfolio.
  6. Trading

    Understanding How Dividends Affect Option Prices

    Learn how the distribution of dividends on stocks impacts the price of call and put options, and understand how the ex-dividend date affects options.
  7. Trading

    Breaking Down The Binomial Model To Value An Option

    Find out how to carve your way into this valuation model niche.
  8. Markets

    How & Why Interest Rates Affect Options

    The Fed is expected to change interest rates soon. We explain how a change in interest rates impacts option valuations.
  9. Trading

    Sensitivity Analysis For Black-Scholes Pricing Model

    Trading options requires complex calculations, based on multiple parameters. Which factors impact option prices the most?
  10. Trading

    Stock Options: What's Price Got To Do With It?

    A thorough understanding of risk is essential in options trading. So is knowing the factors that affect option price.
Trading Center