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Came across the following marketing material where the company called BISAM (FactSet) aka FinAnalytica (?) has developed following fat-tailed volatility model:

$$ r_{t} = \mu + \epsilon_{t} $$ $$ \epsilon_{t} = \sigma_{t} \eta_{t} $$ $$ \sigma_{t}^2 = 0.94 \sigma_{t-1}^2 + 0.06 \epsilon_{t-1}^2 $$

On the other hand, EWMA volatility model takes the form:

$$ \sigma_{t}^2 = 0.94 \sigma_{t-1}^2 + 0.06 r_{t-1}^2 $$

So, BISAM is essentially replacing the term $ r_{t-1}^2 $ with $ \epsilon_{t-1}^2 = (\sigma_{t-1} \eta_{t-1})^2 $.

I was curious, how can that $ \epsilon_{t} $ term could be modelled in order to obtain a fat-tailed model?

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The fat tail features is embedded in the ηt term. In their marketing material (page 4), you will find:

ηt are modelled by a Cognity patented fat-tailed distribution

So basically you don't have a lot of information about this fat tail distribution. You have a lot of models which are more or less related to this one. For example you can think about Filtered Historical Simulations by Barone-Adesi, in which you can fit a long period of returns with a GARCH model (for example 10 years with a GJR-GARCH), and then save your innovations, which contains all the fat tail behaviour. Then when you realize your simulations you draw your innovation from your historical sample.

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Two modeling approaches are commonly used in finance to get a volatility smile and, equivalently, fat tails for the implied returns distribution of e.g. a stock:

  • Assuming a local volatility, i.e. a dependency between the stock price or return and the volatility.
  • Assuming a stochastic volatility (with its own volatility).

What BISAM do is close to the second approach, the variance $\sigma_t^2$ has a deterministic part $0.94 \sigma_{t-1}^2$ and a stochastic part $0.06(\sigma_{t-1}\eta_{t})^2$.

Even if the $\eta_t$ process is a gaussian white noise, you will get a fat-tailed distribution of returns.

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  • $\begingroup$ I was just trying this out in Excel ... when I replace $r_{t-1}^2$ in the standard EWMA model with $(\sigma_{t-1} \eta_{t})^2$ the whole thing goes down to 0. $\eta_{t}$ is obtained using Excel's NORM.INV(RAND(), 0, 0.05). Can you please comment on this? $\endgroup$ – AK88 May 3 '18 at 5:31
  • $\begingroup$ Could you share a subset of your returns data, so that I can have a look? $\endgroup$ – byouness May 3 '18 at 10:11
  • $\begingroup$ I'd love to, but its huge and don't think I can attach it here. You can work S&P500 data from early 2000s (which is available via Yahoo/Google Finance), if you want. $\endgroup$ – AK88 May 3 '18 at 11:50
  • $\begingroup$ If we completely ignore the returns term in the standard EWMA model and replace it with previous volatility of its own, how are we going to take care of current/recent market changes? $\endgroup$ – AK88 May 3 '18 at 12:13
  • $\begingroup$ It seems to me that $\eta_t$must have a mean of 1, not zero, and should be $0<\eta_t<\infty$ $\endgroup$ – Alex C May 4 '18 at 20:20

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