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.

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.

 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
Related Articles
1. Investing

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. Investing

The Volatility Surface Explained

Learn about stock options and the "volatility surface," and discover why it is an important concept in stock options pricing and trading.
3. 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.
4. 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.
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

Breaking Down The Binomial Model To Value An Option

Find out how to carve your way into this valuation model niche.
7. Trading

Dividends, Interest Rates And Their Effect On Stock Options

Learn how analyzing these variables are crucial to knowing when to exercise early.
Frequently Asked Questions
1. Do interest rates increase during a recession?

Learn why interest rates do not rise in a recession; in fact, the opposite happens. Identify the factors that reduce interest ...
2. What is the difference between deflation and disinflation?

Learn what deflation and disinflation are, how supply and demand affect price levels, and the difference between deflation ...
3. What rights do all common shareholders have?

Learn what rights all common shareholders have, and understand the remedies that can be taken if those rights are violated ...
4. What does CHIPS UID mean?

Learn what CHIPS UID stands for and how it facilitates the transfer of funds as the back-end of the ACH network for both ...
Trading Center