Understanding Historical Volatility and How to Calculate It Using Standard Deviation

As covered in my previous blog, historical volatility (HV) measures the extent to which the price of an asset has fluctuated in the past. It is an essential metric for investors and traders to assess risk and make informed decisions.

One of the most common ways to calculate historical volatility is by using the statistical concept of standard deviation. In this blog, I shall break down the process step by step, ensuring clarity even for those new to the subject.

What is Historical Volatility?

Historical volatility quantifies the past price movements of a security and is typically expressed as an annualised percentage in most basic financial analyses. However, in trading, it may be expressed in different time frames, such as even on the hourly frequency. High volatility indicates significant price swings, while low volatility suggests more stable prices. Traders and analysts use historical volatility to:

Estimate potential future price movements.
Price options and other derivatives.
Assess the risk associated with a particular security.

Using Standard Deviation to Calculate Volatility

Standard deviation measures the dispersion of data points (in this case, price returns) around their mean (average). It is a natural choice for measuring volatility because it provides a precise mathematical representation of price fluctuations.

For traders, standard deviation, in simple terms, tells us:

How tightly or widely prices are distributed around the average.
The degree of risk associated with an asset based on past price movements.

Step-by-Step Guide to Calculating Historical Volatility

To calculate historical volatility using standard deviation, follow these steps:

1. Gather Historical Price Data

First, obtain historical price data for the asset you are analysing. Several free data sources, such as https://finance.yahoo.com, from which you can download historical data, are available. Most traders use daily closing prices, but you may also use intraday, weekly, or monthly data, depending on your trading time frame. Usually, the downloaded data will have the OHLC format, which stands for open, high, low, and close prices. Closing prices for whichever time frame you are analysing are often used for this purpose.

Let us assume you are analysing 10 days of closing prices:

Day	Closing Price (£)
1	100
2	102
3	101
4	103
5	105
6	107
7	110
8	108
9	109
10	111

2. Calculate Daily Returns

Next, calculate the daily returns from the price data. Daily returns are the percentage change in the price from one day to the next. Use the formula:

$R_t = \ln\left(\frac{P_t}{P_{t-1}}\right)$

Where:

$R t $ = daily return on day $t$
$P t $ = price on day $t$
$P t - 1 $ = price on day $t - 1$
$ln$ = natural logarithm (used to normalise returns)

The use of natural logarithmic returns—often referred to as logarithmic returns or log returns—is preferred in many cases in finance for mathematical properties such as time-additivity, symmetry in positive and negative returns, and continuous compounding assumptions. Although prices in financial assets cannot go negative (except in futures markets), I prefer to use log returns here as they approximate a normal distribution more closely than simple returns.

3. Compute the Mean of Returns

Once you have daily returns, calculate their average (mean):

$\mu = \frac{\sum_{t=1}^n R_t}{n}$

Where:

$μ$ = mean of the daily returns
$n$ = total number of returns

4. Calculate the Standard Deviation

The standard deviation measures how much the returns deviate from the mean. Use the formula:

$\sigma = \sqrt{\frac{\sum_{t=1}^n (R_t – \mu)^2}{n-1}}$

Where:

$σ$ = standard deviation
$R t $ = daily return on day $t$
$μ$ = mean of the returns
$n$ = number of returns

5. Annualise the Volatility

Daily standard deviation reflects the volatility over one trading day. To annualise it, multiply by the square root of the number of trading days in a year (typically 252 for financial markets):

$HV = \sigma \times \sqrt{252}$

Where:

$H V$ = annualised historical volatility
$σ$ = standard deviation of daily returns
$252$ = approximate number of trading days in a year

Practical Example

Let us say you have 10 days of closing prices for a stock:

Day	Closing Price (£)
1	100
2	102
3	101
4	103
5	105
6	107
7	110
8	108
9	109
10	111

Calculate Daily Returns: Use the formula for daily returns.
Find Mean Return: Compute the average of these returns.
Compute Standard Deviation: Calculate the dispersion of returns.
Annualise Volatility: Multiply the standard deviation by $\sqrt{252}$

This process yields a percentage that represents the historical volatility of the stock.

Key Considerations

Length of Period: Longer periods smooth out short-term noise, providing a broader view of an asset’s volatility. Conversely, shorter periods can highlight recent volatility spikes but may be more sensitive to market anomalies.
Frequency of Data: The choice of data frequency—daily, weekly, or monthly—affects the volatility measure. Daily data provides a detailed, high-frequency view, while monthly data captures broader trends. For most trading applications, daily returns are preferred due to their balance of granularity and applicability.
Assumption of Continuity: Historical volatility relies on the assumption that past price patterns might continue into the future. However, market conditions, geopolitical events, or economic shifts can lead to abrupt changes that historical data cannot predict.
Distribution Assumptions: The calculation assumes that returns follow a particular distribution (e.g., normal or log-normal). While log-normal distributions are more reflective of real-world price behaviour (since prices cannot be negative), no single distribution fully captures the complexities of financial markets.

As a result of using log returns, the return distribution is a log-normal distribution. This has significant trading implications for traders, especially when carrying out risk management.

There is a difference in the way the shape and tail of the log-normal distribution is formed compared to a normal distribution (bell-shaped curve).

In a normal distribution:

The tails decay exponentially, meaning extreme values (very high or very low) are rare.
The kurtosis (measure of “peakedness” and tail behaviour) is fixed at 3.

In a log-normal distribution:

The peak is generally sharper (more pronounced), and the right tail is significantly fatter compared to the normal distribution.
This implies a higher likelihood of extreme positive values, a common feature in financial price data.
The fatter tails account for the occurrence of large price movements that are not well captured by a normal distribution.

This helps explain why I chose a log-normal assumption of price returns while modelling my trading strategies, as it:

Better reflects reality since prices cannot be negative.
Accounts for asymmetry and large upward movements often observed in markets.
Is used in models like the Black-Scholes option pricing model, which assumes that underlying asset prices follow a geometric Brownian motion (implying a log-normal distribution).

While a normal distribution assumption is likely to underestimate the probability of extreme movements (black swans) because of its thin tails.

Market-Specific Dynamics: Volatility differs between markets and asset classes. For instance, equities tend to exhibit higher volatility than bonds, and cryptocurrencies typically show significantly higher volatility than traditional asset classes. It is crucial to consider these dynamics when comparing volatility across markets.
Application in Risk Management: Historical volatility is a key input for risk models, such as value-at-risk (VaR), but it should be used in conjunction with other metrics, such as implied volatility or beta, to build a comprehensive risk management framework.
Adaptation for Different Markets: The trading environment can impact how volatility is perceived. For instance, in emerging markets, where regulatory and economic factors might be less stable, historical volatility might not reflect the full spectrum of potential risks.

Conclusion

Historical volatility is a cornerstone of financial analysis, offering insights into the risk and behaviour of securities. Calculating it using standard deviation is a straightforward yet powerful approach that equips traders and investors with a deeper understanding of market dynamics.

By following the steps outlined above, you can effectively measure and interpret historical volatility, applying this knowledge to optimise your investment strategies. Remember, while historical data is helpful, it is equally important to stay attuned to current market trends and conditions.

Caio Marchesani