The BISAM fat-tailed volatility model vs EWMA volatility model

Question

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?

Answer 1

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.

Answer 2

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.