Fat-Tailed Innovations and EWMA-Based Volatility Models
Summary
The document compares a volatility recursion attributed to BISAM with a standard EWMA form. Both update variance using a weighted prior variance and a squared recent observation, with coefficients shown as 0.94 and 0.06. The key distinction is that the BISAM expression uses the squared residual innovation rather than the squared return. The question is how this construction produces fat tails.
One response says the tail behavior comes from the innovation multiplier, which the cited marketing material describes as following a proprietary fat-tailed distribution. It also points to filtered historical simulation: fit a volatility model, retain its standardized innovations, and sample from those observations when simulating. Another response relates fat tails and volatility smiles to local or stochastic volatility, and argues that even Gaussian noise in the described recursion can produce fat-tailed returns. The document gives no distributional specification or empirical comparison, so it does not establish the exact mechanism or relative performance of the BISAM and EWMA models.
Key ideas
- The compared variance updates differ in whether they use squared returns or squared residual innovations.
- One explanation attributes fat tails to a non-Gaussian innovation distribution.
- Filtered historical simulation can reuse sampled innovations from a fitted volatility model.
- The document also links volatility dependence to fat-tailed returns and volatility smiles.
- No detailed distribution specification or empirical model comparison is provided.
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# The BISAM fat-tailed volatility model vs EWMA volatility model
# 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?
## Answer by DomingoBrown (score 2)
https://quant.stackexchange.com/a/39558
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 by byouness (score 1)
https://quant.stackexchange.com/a/39575
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.