Standard Deviation in Trading: Calculations, Use Cases, Examples and More – Part II

Learn how to calculate standard deviation with Part I.

Standard deviation in trading as a measure of volatility

In trading and finance, it is important to quantify the volatility of an asset. An asset’s volatility, unlike its return or price, is an unobserved variable.

Standard deviation has a special significance in risk management and performance analysis as it is often used as a proxy for the volatility of a security. For example, the well-established blue-chip securities have a lower standard deviation in their returns compared to that of small-cap stocks.

On the other hand, assets like cryptocurrency have a higher standard deviation, as their returns vary widely from their mean.

Moving forward, let us discuss the computation of the annualised volatility of stocks using Python.

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Computing annualised volatility of stocks using Python

Let us now compute and compare the annualized volatility for two Indian stocks namely, ITC and Reliance. We begin with fetching the end of day close price data using the yfinance library for a period of the last 5 years:

import yfinance as yf
import warnings
warnings.filterwarnings('ignore')

# Download the data for ITC and RELIANCE stocks using yahoo finance library
itc_df = yf.download('ITC.NS', period = '5y')[['Adj Close']]
reliance_df = yf.download('RELIANCE.NS', period = '5y')[['Adj Close']]

# Taking a peek at the fetched data
itc_df.tail()

import_lib_download_data.py hosted with ❤ by GitHub

Output:

Date           Adj Close
2021-10-19     245.949997
2021-10-20     246.600006
2021-10-21     244.699997
2021-10-22     236.600006
2021-10-25     234.350006

reliance_df.tail()

show_data.py hosted with ❤ by GitHub

Output:

Date          Adj Close
2021-10-19   2731.850098
2021-10-20   2700.399902
2021-10-21   2622.500000
2021-10-22   2627.399902
2021-10-25   2607.300049

Below, we calculate the daily returns using the pct_change() method and the standard deviation of those returns using the std() method to get the daily volatilities of the two stocks:

# Compute the returns of the two stocks
itc_df['Returns'] = itc_df['Adj Close'].pct_change()
reliance_df['Returns'] = reliance_df['Adj Close'].pct_change()
print(reliance_df[['Adj Close','Returns']])

compute_ret.py hosted with ❤ by GitHub

Output:

Date         Adj Close   Returns
2016-10-25   511.991608       NaN
2016-10-26   508.709717 -0.006410
2016-10-27   506.127686 -0.005076
2016-10-28   509.144104  0.005960
2016-11-01   507.237701 -0.003744
...                 ...       ...
2021-10-19  2731.850098  0.008956
2021-10-20  2700.399902 -0.011512
2021-10-21  2622.500000 -0.028848
2021-10-22  2627.399902  0.001868
2021-10-25  2607.300049 -0.007650

# Compute the standard deviation of the returns using the pandas std() method
daily_sd_itc = itc_df['Returns'].std()
daily_sd_rel = reliance_df['Returns'].std()

reliance_df.dropna(inplace=True)
reliance_df.head()

compute_sd_ret.py hosted with ❤ by GitHub

Output:

Date            Adj Close	 Returns
2016-10-26	508.709717	-0.006410
2016-10-27	506.127686	-0.005076
2016-10-28	509.144104	 0.005960
2016-11-01	507.237701	-0.003744
2016-11-02	494.086243	-0.025928

In general, the volatility of assets is quoted in annual terms. So below, we convert the daily volatilities to annual volatilities by multiplying with the square root of 252 (the number of trading days in a year):

import numpy as np
# Annualized standard deviation
annualized_sd_itc = daily_sd_itc * np.sqrt(252)
annualized_sd_rel = daily_sd_rel * np.sqrt(252)
print(f'The annualized standard deviation of the ITC stock daily returns is: {annualized_sd_itc*100:.2f}%')
print(f'The annualized standard deviation of the Reliance stock daily returns is: {annualized_sd_rel*100:.2f}%')

annualized_sd.py hosted with ❤ by GitHub

Output:

The annualized standard deviation of the ITC stock daily returns is: 27.39%
The annualized standard deviation of the Reliance stock daily returns is: 31.07%

Now we will compute the standard deviation with Bessel's correction. To do this, we provide a ddof parameter to the Numpy std function. Here, ddof means ‘Delta Degrees of Freedom'.

By default, Numpy uses ddof=0 for calculating standard deviation- this is the standard deviation of the population. For calculating the standard deviation of a sample, we give ddof=1, so that in the formula, (n−1) is used as the divisor. Below, we do the same:

# Compute the standard deviation with Bessel's correction
daily_sd_itc_b = itc_df['Returns'].std(ddof=1)
daily_sd_rel_b = reliance_df['Returns'].std(ddof=1)

# Annualized standard deviation with Bessel's correction
annualized_sd_itc_b = daily_sd_itc_b * np.sqrt(252)
annualized_sd_rel_b = daily_sd_rel_b * np.sqrt(252)
print(f'The annualized standard deviation of the ITC stock daily returns with Bessel\'s correction is: {annualized_sd_itc_b*100:.2f}%')
print(f'The annualized standard deviation of the Reliance stock daily returns with Bessel\'s correction is: {annualized_sd_rel_b*100:.2f}%')

Bessel's_corr.py hosted with ❤ by GitHub

Output:

The annualized standard deviation of the ITC stock daily returns with Bessel's correction is: 27.39%

The annualized standard deviation of the Reliance stock daily returns with Bessel's correction is: 31.07%

Thus, we can observe that, as the sample size is very large, Bessel's correction does not have much impact on the obtained values of standard deviation. In addition, based on the given data, we can say that the Reliance stock is more volatile compared to the ITC stock.

Note: The purpose of this illustration is to show how standard deviation is used in the context of the financial markets, in a highly simplified manner. There are factors such as rolling statistics (outside the scope of this write-up) that should be explored when using these concepts in strategy implementation.

Stay tuned for the next installment to learn about z-score.

Author: Chainika Thakar (Originally written by Ashutosh Dave and Udisha Alok)

Originally posted on QuantInsti blog.