I have a time series, let's denote it as $X_t$ (a series of prices). We exponentially smooth the data to obtain another time series, an exponential moving average: $EMA_t$. My question is how do I calculate the moving variance $\sigma^2_t$ of $EMA_t$? I'm pretty sure the standard sample variance $\frac{1}{N}\sum_i (y_i - \hat{\mu})^2$ is not the way to go about calculating the rolling variance. I'm doing this in excel, so I have an actual list of prices that I exponentially smoothed and would like to calculate a rolling variance for.
From here we have that the basic form of exponential smoothing is $$s_t = \alpha x_t + (1-\alpha) s_{t-1} $$ Is there a way to manipulate the recurrence such that I can obtain the moving variance?
Does anyone have any ideas? Could it be related to finding the variance of an AR(1) model?
I think the answer is $EMSD_t = \alpha (s_t - x_t)^2+ (1-\alpha)s_{t-1}^2$.