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I am trying to apply a volatility strategy. I am reading a paper where the authors defined the volatility as: "Exponential Weighted Volatility of returns with a 1-year window and 3-month half-life"

I am having a hard time understanding the mathematical formula underlying it. The 1-year window part is easily understood as a summation of weighted square return deviation up to 12 months back. I think that the 3-month half-life is used for the weights but cannot figure out the exact mathematical representation. Any help on this is appreciated.

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The Exponentially Weighted Moving Average (EWMA for short) is characterized my the size of the lookback window $N$ and the decay parameter $\lambda$.

The corresponding volatility forecast is then given by: $$ \sigma_t^2 = \sum_{k = 0}^N \lambda^k x_{t-k}^2 $$

Sometimes the above expression is normed such that the sum of the weights is equal to one. However, for large $N$ this makes no difference.

Coming to your question, instead of providing $\lambda$ the half-life $\tau$ can be provided as well. The half-life is the time lag at which the exponential weights decay by one half, i.e. $$ \lambda^\tau = \frac 12 \iff \tau = - \frac{\ln2}{\ln \lambda} \iff \lambda = \biggl(\frac 12\biggr)^{\frac 1\tau}. $$

In your case $\tau = \frac 14$ which means that after 3 months the weights in the EWMA are less or equal than $\frac 12$. The corresponding value for $\lambda$ is then given by $\lambda = \bigl(\frac 12\bigr)^{\frac 1\tau} = \frac {1}{16}$.

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  • $\begingroup$ Thank you, i have a question regarding that 2 in the half life formula, is it in 'years' unit? What i mean to say is, do i use half life = 0.25 or 3? $\endgroup$
    – Dhruv Mahajan
    Jun 19, 2019 at 12:12
  • $\begingroup$ Usually it is measured in years, I updated my answer. $\endgroup$
    – Cettt
    Jun 19, 2019 at 12:14
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    $\begingroup$ @Cettt: Nice answer but two things, neither of which are a big deal. 1) the EMA is usually defined to go all the way back to the beginning of the data so that $N = \infty$. 2) If the OP wants to estimate the volatility of returns, and the mean is not assumed to be zero, then some mean estimate should be subtracted off of $x_{t}^2$. This adds complications ( what estimate should one use ) so hopefully it can be assumed to be zero. $\endgroup$
    – mark leeds
    Jun 21, 2019 at 19:31
  • $\begingroup$ very good explanation of EWMA $\endgroup$
    – Stephen Ge
    Nov 28, 2022 at 2:46

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