Volatility in Lognormal Distribution
To calculate the volatility of continuously compounded returns, first calculate the continuously compounded returns. Then calculate the standard deviation of these returns and annualize the standard deviation using $$\sigma ^{2}T$$, where $$\sigma ^{2}$$ is the variance and $$T$$ is the number of periods.
So give it a shot. Suppose you have the following price information:
| Date | Price |
|---------------|-------|
| 13 April 2014 | $10.00 |
| 14 April 2014 | $9.50 |
| 15 April 2014 | $9.75 |
| 16 April 2014 | $9.25 |
What is your first step to calculate the volatility of the daily price data in annualized form assuming 250 trading days?
Incorrect.
Volatility is based on returns and not prices. You should first calculate continuously compounded returns for each day.
So what is the continuously compounded return for April 14, 2014?
Incorrect. This is the holding period return, and here you need the continuously compounded return.
Repeat this same calculation for the other returns, and you'll find the remaining continuously compounded returns.
To April 15, 2014: $$ ln(\frac{9.75}{9.50}) = 0.026 $$.
To April 16, 2014: $$ ln(\frac{9.25}{9.75}) = -0.053 $$.
Now see if you can find the sample standard deviation of these returns and annualize. What answer do you get?
Incorrect.
This is the sample standard deviation, but it has not been annualized.
Yes!
Since the average of these three is -0.026, the sample standard deviation is
$$ \displaystyle \sqrt{\frac{(-0.051+0.026)^2 + (0.026 + 0.026)^2 + (-0.053 + 0.026)^2}{3-1}} $$
$$ \displaystyle \approx 0.045 = 4.5 \% $$
To annualize under the assumption of 250 trading days in a year, you take:
$$\displaystyle 0.045\sqrt{250} \approx 0.712 = 71.2 \% $$
which means that annualized volatility in this example is 71.2%.
Incorrect.
Be sure that, when you annualize, you take the square root of the number of days.
That's right!
Good work!
To summarize:
[[summary]]
Calculating __volatility__ of continuously compounded returns is important for option pricing models, such as Black-Scholes-Merton. If prices are assumed to be the result of continuously compounded returns, then the natural logarithm of the series may be approximately normally distributed. In this case, the price is a __lognormal distribution__, a continuous probability distribution of a random variable whose logarithm is normally distributed. Volatility of such distributions is typically reported as an annualized standard deviation of continuously compounded returns.
The continuously compounded return to April 14, 2014 is
$$\displaystyle ln(\frac{9.50}{10}) = -0.051 $$.
Finding the standard deviation of the prices
Calculating continuously compounded returns for each day
-0.050
-0.051
4.5%
71.2%
1,125.2%
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