The simple answer is the standard deviation of periodic returns. This video takes some sample data for closing prices of a stock and demonstrates how volatility is calculated in Excel.

In finance, such as for price series, usually log returns are used, where log is the natural logarithm.

<iframe width="600" height="450" src="http://www.youtube.com/embed/5_rxNA-nn7k" frameborder="0" allowfullscreen></iframe>

What is Volatility?

Volatility and VaR can be scaled using the square root of time rule.  According to this rule, if the fluctuations in a stochastic process are independent of each other, then the volatility will increase by square root of time. It provides exact volatilities if the volatilities are based on lognormal returns. In case of simple returns the scaled volatilities are approximate.

Let's say the monthly volatility is 5%, then the annual volatility will be 0.05*Sqrt(12) = 17.32%.

<iframe width="600" height="450" src="http://www.youtube.com/embed/SlQSFjJATG4" frameborder="0" allowfullscreen></iframe>

What is the Square Root Rule?

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 100 to base 10 is 2, because 100 is 10 to the power 2: 1000 = 10 × 10  = 10<sup>3</sup>. More generally, if x = b<sup>y</sup>, then y is the logarithm of x to base b, and is written y = log<sub>b</sub>(x), so log<sub>10</sub>(100) = 2.

The inverse of a logarithmic function is an exponential functions. 

Logarithms play a significant role in quantitative finance. If we use a base of natural e, we can compute continuously compounded returns.

The first video below provides an intuitive explanation of the concept of logarithms.

<iframe width="600" height="450" src="http://www.youtube.com/embed/djR8DkQ_UU0" frameborder="0" allowfullscreen></iframe>

The video below discusses why we use log returns in finance.

<iframe width="600" height="450" src="http://www.youtube.com/embed/PtoUlt3V0CI" frameborder="0" allowfullscreen></iframe>

Why Use Lognormal Returns in Finance (Stock Prices)?

We know that the <a href="https://financetrain.com/what-is-the-square-root-rule/" title="What is the Square Root Rule?">square root rule</a> can be used to scale volatility with time. This rule assumes that the returns are independent and identically distributed. However, this assumption is not very realistic.  

This video illustrates a scaling factor that adjusts the square root rule for for <a href="https://financetrain.com/serial-correlation-autocorrelation/" title="What is Serial Correlation (Autocorrelation)?">autocorrelation</a>.

<iframe width="600" height="450" src="http://www.youtube.com/embed/Ms3uu9TKcgA" frameborder="0" allowfullscreen></iframe>

This video is developed by David from Bionic Turtle.

How to Scale Autocorrelated Returns?

This video provides clarity about the confusion over the stock returns.  The expected return of a stock is ambiguous because, if we assume returns are normal, then price levels are lognormal. In which case, the mean does not equal the median future stock price.

Consider these two questions:

1. What is the average return per month to realize a return of 20% over a period of 1 year.
2. What is the stock's return in an average month over a period of 1 year.  

In the first question, we are actually looking at geometric mean, while in the second question we are looking at arithmetic mean.

<iframe width="600" height="450" src="http://www.youtube.com/embed/9YLIiWb_NYk" frameborder="0" allowfullscreen></iframe>

This video is developed by David from Bionic Turtle.

Arithmetic Vs. Geometric Stock Returns

Extreme value theory (EVT) aims to remedy a deficiency with <a href="https://financetrain.com/value-at-risk-var/" title="Value at Risk (VaR)">value at risk</a> (i.e., it gives no information about losses that exceed the VaR) and glaring weakness of delta normal value at risk (VaR): the dreaded-fat tails. The key is the idea that the tail has it's own "child" distribution.

This video explains the extreme value theory.

<iframe width="600" height="450" src="http://www.youtube.com/embed/o-cpu1IH3tM" frameborder="0" allowfullscreen></iframe>

This video is developed by David from Bionic Turtle.


Extreme Value Theory

Introduction to Quantitative Finance

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Introduction to Quantitative Finance

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