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I read somewhere that the decay factor is (1-lamba)*lamba^t where t is first return, second return, third return, ...

I also found this formula which I have difficulty to understand:

volatility of an asset with decay factor

How do I calculate the volatility of my asset taking into account my 0.97 decay factor? I have the time series of daily prices.

Normally to get volatility I would get the average daily return, then calculate the variance by summing all the (return-average return)^2 and dividing by N.

I would then square the variance and multiply by square(25) to have a volatility for a period of 1 month.

At what stage am I supposed to incorporate the decay factor?

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Effectively, you take your series of prices and transform those to returns as you would do in your standard approach, as well.

step   return date  weight
1      2020-03-26   0.97
2      2020-03-25   0.97 * 0.97
3      2020-03-24   0.97 * 0.97 * 0.97
...
K      T-K+1        097^K

For $K$ sufficiently large, the sum of the weights $\sum_i \lambda^i$ will converge to $\frac{1}{1-\lambda}$, hence the prefactor in your formula.

In order to calculate your volatility, you do not sum the squared returns and divide by $N$, as you would do normally, but you compute the weighted sum of your squared returns, normalised by $1-\lambda$. You may then calculate the square root to arrive at your $\sigma_t$.

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