# Calendar Days versus Trading Days

The “calendar days” component in the formula in Table 7-5 is often debated. The question is, “Should the formula use calendar days or trading days?” Those in favor of trading days argue that volatility, that is, stock-price change, can only happen on trading days. Others counter that calendar days better reflect the actual amount of time until expiration. The answer is: In most cases, either can be used without much impact on the result.

The conversion formula uses square root of time in years, so the important question is whether calendar days or trading days best approximates time in years. It can be argued, generally, that it does not matter.Any percent of a full year is the same regardless of the number of days in a year. Choosing 252 (trading days) versus 365 (calendar days) for days per year for price-range estimates using volatility becomes an issue only when the time period is short. What is a “short” time period? Two examples follow that shed light on this issue.

First, consider a two-month time period in which there are 61 calendar days and 43 trading days. Also assume a stock price of 78.50 and 35 percent volatility. Calendar days are used to calculate a standard deviation for the period as follows: 61 calendar days divided by 365 calendar days in a year is 0.1671, the square root of which is 0.4087. The volatility for the period, therefore, is 14.3 percent (0.35  0.4087). And for a stock trading at 78.50, one standard deviation is 11.23 (78.50  0.143).

Trading days are used to calculate a standard deviation for the period as follows: 43 trading days divided by 252 trading days in a year is 0.1706, the square root of which is 0.4130. The volatility for the period, therefore, is 14.5 percent (0.35  .4130). And for a stock trading at 78.50, one standard deviation is 11.38 (78.50  0.145).The difference between using calendar days and trading days is 15 cents.

For a stock price of 78.50 and a period of two months, this difference is not significant.Second, consider a three-day time period, again assuming a stock price of 78.50 and volatility of 35 percent. Using calendar days, 3 divided by 365 calendar days in a year is 0.0082, the square root of which is 0.0906. The volatility for the period, therefore, is 3.2 percent (0.35  0.0906). And for a stock trading at 78.50, one standard deviation is 2.51 (78.50  0.032).

Using trading days, 3 divided by 252 trading days in a year is 0.0119,the square root of which is 0.1091. The volatility for the period, therefore,is 3.8 percent (0.35  0.1091). And for a stock trading at 78.50,one standard deviation is 2.98 (78.50  0.038).The difference between using calendar days and trading days is 47 cents (2.51 versus 2.98). This is approximately a 17 percent difference and arguably significant.

How much of a concern should the difference between using calendar days and trading days be to traders? For a two-month time period, given a stock price of 78.50, most traders would not consider 15 cents to be significant. For the three-day period, the difference of 47 cents might be significant depending on how often a trader uses strategies targeted at three days.In general, the answer also partly depends on how accessible the necessary information is.

Most traders have easy access to the number of calendar days to expiration because brokers supply it. In contrast, the number of trading days is more difficult to find and time-consuming to calculate. Many traders therefore use calendar days when converting annual volatility to shorter time periods because it is easier, and it usually does not make much difference.

The focus now will shift from volatility as it relates to stock-price movements to volatility as it relates to option prices. Remember, from Chapter 2, that volatility is one of the six components that influence option prices.Implied Volatility:Implied volatility is the volatility percentage that justifies the market price of an option. In other words, it is the volatility percentage that returns the option’s market price as the theoretical value.

This concept is best explained with an example.Consider Gary, who uses Op-Eval Pro to estimate the theoretical value of an XYZ March 70 Call. Figure 7-6 shows a Single Option Calculator screen from Op-Eval Pro with Gary’s inputs: current stock price of 68.00, strike price of 70, no dividend, interest rate of 4 percent,and 75 days to expiration. Gary chose a volatility of 26 percent because that percentage was the historic volatility based on the 30 most recent daily closing stock prices. Given Gary’s inputs, Op-Eval Pro calculates a value of 2.57 for the XYZ March 70 Call.