Annualized Volatility with simple returns

Question

I am aware of the way to annualize a volatility using log returns (stdev(daily log returns) * sqrt(252)).

But how can I do the same if I have a time series of simple returns (price_t - price _t-1) / price_t ?

Answer 1

Depends on what you're trying to do.

Log-normal model

Usually, you'd compute the Vol of Log-returns if you're trying to calibrate a Log-normal model, such as the Geometric-Brownian-Motion model for the stock price under the real-world probability measure:

$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$

If you need to calibrate a model such as the above and if you were given computed regular returns instead of log-returns, you can just turn the regular returns into log-returns as follows:

$$LogReturn :=ln\left(\frac{p_{t_i}}{p_{t_{i-1}}}\right):=x$$

$$RegularReturn=\frac{p_{t_i} - p_{t_{i-1}}}{p_{t_i-1}}:=y$$

=>

$$y=\frac{p_{t_i}}{p_{t_{i-1}}}-1=e^x-1$$

$$y+1=e^x$$

$$ln\left(y+1\right)=x$$

So basically, add 1 to all your regular returns and take log of the result, and you convert your regular returns to log returns (without having to recover the original data). Then proceed as you would with log returns.

Normal model

If your model is normal, as follows:

$$ dS_t = \mu dt + \sigma dW_t $$

Then:

$$ S_t - S_0 \sim N(\mu t, \sigma \sqrt{t}) => \frac{S_t - S_0}{S_0} \sim N(\frac{\mu t}{S_0}, \sigma \frac{\sqrt{t}}{S_0}) $$

Therefore if you need to calibrate such model, you can just compute your returns as $p_{t_i} - p_{t_{i-1}}$ then compute the standard deviation of these differences and then annualize it by multiplying through by $\sqrt{252}$.

If you compute your returns as $\frac{p_{t_i} - p_{t_{i-1}}}{p_{t_i-1}}$ and then compute the standard deviation of these, you need to annualize your computed standard deviation by multiplying through by $S_0\sqrt{252}$, whereby for $S_0$ you could take the average price over your time series.

In summary: it depends on what you're trying to achieve with computing the volatility of your returns.

Answer 2

In the same way. Just multiply the daily standard deviation for the square root of the trading days (i.e. 252).