Black-Scholes Inputs (Parameters)
6 Inputs of the Black-Scholes Option Pricing Model
There are 6 basic inputs (parameters) to set when pricing an option using the Black-Scholes model. They are the following:
- Underlying price
- Strike price
- Time to expiration
- Risk-free interest rate
- Dividend yield
Below you can find more details and notes concerning individual inputs.
Underlying Price Input
Underlying price is the price at which the underlying security is trading on the market at the moment you are doing the option pricing. For example, if you are pricing an option on J.P. Morgan (JPM) stock and the stock is trading at 44.50 at the time you are doing the pricing, you enter 44.50 as underlying price.
Strike Price Input
Strike price, also called exercise price, is the price at which you would buy (or sell) the underlying security if you choose to exercise the call (or put) option. It is one of the fixed specifications of each particular option contract and it does not change during the life of the option.
Time to Expiration Input
Time to expiration is the time between the moment you are doing the pricing and the expiration of the option. Like strike price, expiration date is a fixed characteristic of every individual option. The expiration date does not change during the life of the option, but as the time passes, the time left to expiration (the number that enters the Black-Scholes model as an input) decreases. Time to expiration enter the calculations as percentage of year, but most software (including the Macroption Black-Scholes Calculator) enables you to enter today’s date and expiration date (and time if you need to be more precise) and converts it to percentage of year automatically. You may also decide if you want to measure time to expiration in calendar days or trading days.
The volatility input, measured in percent per year, is how much you generally expect the underlying security to move during the life of the option. For people new to option pricing, the volatility concept may be a bit complicated at first, before you fully understand what the number really represents (standard deviation of returns). If you are not much familiar with volatility, hopefully you will find answers to your questions on the main volatility page.
Also note that volatility is probably the one Black-Scholes input that is the hardest to estimate (and at the same time it can have huge effect on the resulting option prices). Two common ways of estimating volatility are:
- By looking at the historical volatility of the underlying asset. You can assume that the underlying asset will have similar volatility going forward (which sometimes hasn’t) or you can adjust the historical volatility by your beliefs about the future (for example make the volatility input slightly higher than the historical volatility you have calculated, because you expect something in the future to make the price move more – for example earnings announcement of the underlying company). It is discusses in detail in chapter 8 of the calculator PDF guide.
- By looking at the implied volatility of similar options (options with other strike prices or expiration dates on the same underlying or options on related securities, for example options on other stocks in the same sector).
Risk-Free Interest Rate Input
Like volatility, risk free interest rate is also measured in percent per year. For a particular trader it should be the rate at which you can deposit or borrow cash over the life of the option (the interest rate tenor should match the time to expiration). The interest rate input is not that important when interest rates are low (like now), but it can get very important when they are high (the importance depends on a particular option’s strike price, type, and time to expiration).
Dividend Yield Input
Dividend yield was not among the inputs in the original version of the Black-Scholes model, but was added soon as an expansion.
Here you can see more details about dividend treatment in the Black-Scholes model and to the respective papers by Black, Scholes, and Merton.