Options Pricing: Black-Scholes Model
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).
|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|
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