How Is Historical Volatility Calculated?

Historical volatility quantifies an asset’s price fluctuation over a specific past period. It provides insight into how much an asset’s price has moved, serving as an indicator of past price behavior. This measurement helps market participants understand the degree of price dispersion an asset has experienced. It is commonly used to assess the historical risk associated with an investment, reflecting the extent of its past price swings. While it does not predict future movements, historical volatility offers a valuable perspective on an asset’s typical range of price changes.

Key Concepts and Data Preparation

The calculation of historical volatility begins with collecting specific historical price data for the asset under analysis. Typically, daily closing prices are used, forming a consistent time series over a defined period. This consistent dataset is fundamental, as any gaps or inconsistencies could distort the volatility measurement. The choice of the “lookback period,” such as 30, 60, or 252 days, determines the length of this historical data series and directly influences the resulting volatility figure.

For accurate volatility calculations, financial professionals often prefer using logarithmic (log) returns instead of simple percentage returns. Log returns are advantageous because they account for continuous compounding, which better reflects how investments grow over time. They also exhibit better statistical properties, such as additivity over time, meaning that the log return over multiple periods is simply the sum of the log returns for each sub-period. This characteristic makes them particularly suitable for financial modeling and analysis.

The formula for calculating a daily logarithmic return for a given day is the natural logarithm (ln) of the current day’s closing price divided by the previous day’s closing price. This mathematical transformation helps normalize the return distribution, making it more amenable to statistical analysis. For instance, if a stock price moves from $100 to $101, the log return would be ln(101/100). This approach provides a more accurate representation of price changes, especially when dealing with large fluctuations.

A core statistical concept underlying historical volatility is standard deviation. Standard deviation measures the dispersion or spread of a set of data points around their average value. In the context of financial markets, a higher standard deviation indicates that an asset’s price has historically exhibited larger swings from its average, implying greater volatility. Conversely, a lower standard deviation suggests more stable prices with less deviation from the average.

Therefore, historical volatility is essentially the standard deviation of an asset’s past returns. It quantifies the degree to which an asset’s price has deviated from its average over a given period. This measure provides a numerical representation of an asset’s price instability, allowing for a standardized comparison of risk across different investments. The lookback period chosen for this calculation significantly impacts the outcome, as shorter periods might capture recent market sentiment more acutely, while longer periods offer a broader historical perspective.

The Core Calculation Steps

Calculating historical volatility involves a series of sequential mathematical operations, starting with raw price data. The first step requires gathering a consistent series of historical closing prices for the asset over the chosen lookback period, such as five consecutive days.

| Day | Closing Price ($) |
| :– | :—————- |
| 1 | 100 |
| 2 | 102 |
| 3 | 101 |
| 4 | 103 |
| 5 | 105 |
| 6 | 104 |

• Calculate the daily logarithmic returns for each period. This is achieved by taking the natural logarithm (ln) of the current day’s closing price divided by the previous day’s closing price. For example, using the provided data, the return for Day 2 is ln(102 / 100) ≈ 0.0198, for Day 3 is ln(101 / 102) ≈ -0.0098, for Day 4 is ln(103 / 101) ≈ 0.0196, for Day 5 is ln(105 / 103) ≈ 0.0192, and for Day 6 is ln(104 / 105) ≈ -0.0095.
• Calculate the average (mean) of these daily log returns. This involves summing all the calculated log returns and then dividing that sum by the total number of returns, denoted as ‘N’. For the example, the sum is approximately 0.0393, and the mean is 0.0393 / 5 = 0.00786.
• Determine the deviation of each individual log return from this calculated mean. This is done by subtracting the mean return from each daily log return. This quantifies how much each daily return varies from the average. For example, Day 2 Deviation: 0.0198 – 0.00786 = 0.01194.
• Square each of these deviations. Squaring eliminates negative values, ensuring all deviations contribute positively to dispersion, and gives greater weight to larger deviations. For example, Day 2 Squared Deviation: (0.01194)^2 ≈ 0.00014256.
• Sum all the squared deviations obtained from the previous step. This sum represents the total squared difference of all returns from the mean, an intermediate step towards calculating the variance. For the example, the sum is approximately 0.00102223.
• Calculate the variance. Variance is determined by dividing the sum of the squared deviations by (N-1), where N is the number of log returns. Using (N-1) is a statistical adjustment for sample variance, providing a more accurate estimate. For the example, the variance is 0.00102223 / (5 – 1) ≈ 0.00025556.
• Calculate the daily historical volatility by taking the square root of the variance. This brings the measure back to the same units as the original returns, making it intuitive to interpret as a percentage. For the example, the daily historical volatility is the square root of 0.00025556, which is approximately 0.015986, or 1.60%.

Annualizing the Result

Once the daily historical volatility has been calculated, it is common practice to annualize this figure. Annualizing volatility converts the daily measure into an equivalent annual rate, which allows for easier comparison across different assets and investment horizons. Financial markets typically quote volatility on an annual basis, making this conversion a standard convention for reporting and analysis.

The annualization formula is straightforward: the daily volatility is multiplied by the square root of the number of trading days in a year. This scaling factor accounts for the fact that volatility increases with the square root of time, assuming price movements are random and independent over time. This approach allows a short-term volatility measure to be projected over a longer period.

The number of trading days in a year can vary slightly depending on the market and the specific conventions used. For instance, in the United States equity markets, common conventions for the number of trading days typically range from 250 to 252. This figure accounts for weekends and observed market holidays throughout the year. While 252 trading days is a frequently cited number, some calculations may use 250 or 251.

For example, if the calculated daily historical volatility is 1.60%, and using 252 trading days for annualization, the annualized volatility would be 0.0160 √(252). This calculation results in an annualized figure, which in this case would be approximately 0.0160 15.87 ≈ 0.2539, or 25.39%. This means that, based on its past daily movements, the asset would be expected to fluctuate by around 25.39% over the course of a year.

The annualized figure provides a broader perspective on an asset’s expected price swings over a full year. A higher annualized percentage indicates that the asset has historically experienced greater price variability over a 12-month period. This metric is a crucial input for various financial models, including option pricing, and assists investors in assessing the overall risk profile of an investment.