In the option market, historical volatility is calculated as the standard deviation of daily returns on the underlying asset over a specified period of time. In practice, traders and analysts use an annual expression of volatility, just as interest rates are always measured in annual terms.
The natural logarithm of the price is used to calculate daily interest rates.
Example. The share is trading at $100, and in one day the price rises to $102. Simple return for one day is defined as
102/100 – 1 = 2%
Let's say that then the price drops back to $100. The price will fall in simple percentages:
100/102 – 1 = –1,96%
The disadvantage of this calculation method is that percentage changes cannot be added to each other. If the share price starts at $100 and ends at the same level, the price change is 0%, not 2% - 1.96% = 0.04%. Using natural logarithms solves this problem:
Ln(102/100) + Ln(100/102) = 1.98% - 1.98% = 0
The table below illustrates how historical volatility is calculated.
Day |
Price |
Price change |
Deviation |
Deviation^2 |
0 |
500 |
- |
||
1 |
508 |
1,587% |
1,370% |
0,019% |
2 |
492 |
-3,200% |
-3,418% |
0,117% |
3 |
498 |
1,212% |
0,995% |
0,010% |
4 |
489 |
-1,824% |
-2,041% |
0,042% |
5 |
502 |
2,624% |
2,406% |
0,058% |
6 |
507 |
0,991% |
0,773% |
0,006% |
7 |
500 |
-1,390% |
-1,608% |
0,026% |
8 |
502 |
0,399% |
0,182% |
0,000% |
9 |
499 |
-0,599% |
-0,817% |
0,007% |
10 |
511 |
2,376% |
2,159% |
0,047% |
Average = |
0,218% |
Total = |
0,33% |
Historical volatility calculation
Given the average return for a day was 0,22%, we find the sum of squares of deviations equal to 0,33 %. Therefore, the variance:
Variance = Sum of Deviation^2 / (# of Days – 1)
Variance = 0.33%/(10-1) = 0.0033/9 = 0.000367 = 0.0367%
Volatility is defined as the standard deviation of the daily returns of an asset, which is the square root of the variance:
Volatility σ = SQRT (Variance) = SQRT (0.0367%)
Volatility σ = 0.0192 = 1.92%
Thus, the daily volatility of the stock is equal to 1.92%. This indicates that the share has changed on average by 1.92% during one day, both up and down. In other words, the daily share price fluctuations averaged 1.92% per day. However, among option traders it is common to use an annual expression. A year consists of 252 trading days, so to find the annual value of the share's volatility you need to multiply the daily volatility by the square root of 252:
Annual volatility = 1,92% x SQRT (252) = 30,4%
Option traders often compare the realized volatility of the underlying asset during previous reporting periods with the implied volatility of the options of that asset. If the annual realized volatility on the date of quarterly financial results is usually 100%, i.e. 6.3% per day, and the implied volatility of ATM option is at 40%, or 2.52% per day, then a trader can:
- Buy at-the-money options
- hedge the delta to zero
If after the publication of the report, the share continues to show volatility of 100% per annum, the trader will make a big profit (with a relatively low theta) by hedging the delta.