# The BISAM fat-tailed volatility model vs EWMA volatility model

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?

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.

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.

• 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? – AK88 May 3 '18 at 5:31
• Could you share a subset of your returns data, so that I can have a look? – byouness May 3 '18 at 10:11
• 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. – AK88 May 3 '18 at 11:50
• 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? – AK88 May 3 '18 at 12:13
• It seems to me that $\eta_t$must have a mean of 1, not zero, and should be $0<\eta_t<\infty$ – Alex C May 4 '18 at 20:20