The Black-Scholes option pricing formula can't be deconstructed to determine a direct formula for implied volatility. However, if you know the option's price and all the remaining parameters (underlying price, strike price, interest rate, dividend yield, and time to expiration), you can use the Goal Seek feature in Excel to find it. This page explains how to do it in the Black-Scholes Calculator (but the logic is the same if you do it on your own and prepare all the Black-Scholes model formulas yourself).
I will illustrate the Excel calculation of implied volatility step-by-step on the example below.
You want to find implied volatility of a call option with strike price of 55 and 18 calendar days to expiration. The risk free interest rate is 1%; the underlying stock's continuously compounded dividend yield is 2%. The underlying stock is currently trading at 53.20 and the option is trading at 1.40.
Setting the Input Parameters
First, you must set all the parameters that enter option price calculation:
- Enter 53.20 in cell C4 (Underlying Price)
- Enter 55 in cell C6 (Strike Price)
- Cell C8 contains volatility, which you don't know. Just enter something (for example 50%).
- Enter 1% in cell C10 (Interest Rate)
- Enter 2% in cell C12 (Dividend Yield) – if the underlying pays no dividend, enter zero or leave this cell blank.
- Enter 18 in cell C19 (Time to Expiration) and select "Calendar days" in the combo in cell C20. If you don't know the number of days, but you know the expiration date of the option, enter the dates in cells C15 and C17 (if both these cells contain numbers, they will override the input in cell C19). You may also enter exact times in cells C16 and C18 if you want to be very precise.
Now you have entered all the parameters and the resulting option price appears in cell H4 (or H6 if you are working with a put option). Using the values in our example, it is 1.59 dollars per share.
Unless you were very lucky, it is not equal to the actual price at which the option is trading at the moment (in our example 1.40). The reason is the volatility parameter, where you have entered a number you just guessed.
Note: Do not type any numbers in cells H4 and H6 (the resulting option prices). These cells contain formulas and if you overwrite them, the spreadsheet will not work correctly.
Trial and Error Approach
Now you can try to find the implied volatility by trial and error by entering different values in cell C8. If the real option price is lower than your result (as in our example), try lower volatility, and vice versa. For example, you can try to enter 45% into cell C8 and get option price of 1.36. So you try 47% and get 1.45... and so on. As soon you get close enough to the real option price (depending on your desired level of accuracy), you are done. The implied volatility is in cell C8.
Using Excel Goal Seek
The Goal Seek feature in Excel does exactly the same thing, only the computer is able to perform this trial and error exercise in split of a second and get a very accurate result immediately.
Once you have the input parameters set, go to Excel main menu and select Data, Data Tools, What-If Analysis, Goal Seek (in Excel 2010 – the path may be slightly different in other versions).
The Goal Seek window pops up and asks you to enter three inputs:
- "Set cell:" – the cell where the resulting option price is calculated – enter H4 if you are trying to find implied volatility of a call (our example), or H6 for a put.
- "To value:" – the option's price. In our example enter 1.40.
- "By changing cell:" – the cell which contains the input that you want to find – in our example the cell with volatility input, C8.
Now press OK and the desired implied volatility appears in cell C8 (45.83% in our example). At the same time, the option‘s price (1.40 in our example) should appear in cell H4 (or H6 if it was a put). You can also see the option's delta, gamma, theta, vega, and rho to the right of option's price in cells J4 to N4 (or J6 to N6).
This tutorial is part of Black-Scholes Calculator user guide. You can see more information about all features, calculations, and guide contents here.