Gaussianizing Asset Returns with Volatility Models and Factor Models

# Creating a standard Gaussian model on stock market data


# Creating a standard Gaussian model on stock market data












One of the big problems in creating good statistical models in the stock market is because of the long tails that deviate from Gauss' [regular] bell model, is there a way to create a synthetic Gauss bell on market data, by a random walk model that buys long or short [each time] And so it balances the tails completely [after all: if a person enters the market both long and short all the way, he will find himself with a span of zero at the end of the road, and if so a random walk should create a zero span with a standard Gaussian model] I Do not know if my logic is correct, I would love any answer or comment from the great experts here

Thank you

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/55728

I think I understand where you are going, please correct me if I'm wrong. I also think whatever you could do to transform the returns to Gaussian will be very complex or not really useful. In short, I'm not convinced this will be a fruitful approach.

If you have a sample of returns you can always apply a scaling to individual observations to make them normal. In the plot below, take every point that's for from the line and scale it so that it's closer to the line, repeat until your convinced the result is normal. I'm unconvinced this is a good idea. If you keep the observations in order you would at least not lose some forms of auto-correlation.

Assuming (big assumption) that returns are drawn from a mixture of normal $N(0, \sigma)$ distributions a better approach would be to model the volatility of the returns using GARCH and scale the returns using the inverse of the predicted volatility. If the model is correctly specified the returns would now be normal. This has two drawbacks, one model specific, one more fundamental:

- GARCH doesn't forecast upward volatility shocks, you can only now-cast which is a limitation for investing;

- The existence of a simple model that forecasts volatility is unlikely. It will almost be certainly be more complicated than what you're trying to achieve right now.

At least, it has a stronger theoretical basis than the scaling described above.

If we drop the assumption of returns coming from a mixture, you would need another model, presumably a lot more complex. I think Cont, R. (2001) Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1, 223-236. is the standard reference for stylized facts you would want to account for.

## Transforming the data to Gaussian

The plot above is created in R with:

```
set.seed(1L)
returns <- rt(1:1e3, df = 10)
qqnorm(returns)
qqline(returns)
```

### Returns normalized by force

These returns can be made Gaussian as follows:

```
# Don't do this
scaling <- qnorm((1:100) / 101)[order(returns)] / returns
qqnorm(returns * scaling)
qqline(returns * scaling)
shapiro.test(returns * scaling)
```

Results in

```
    Shapiro-Wilk normality test

data:  returns * scaling
W = 0.99774, p-value = 0.9999 # Higher p-value => more evidence of normality
```

This forces `returns * scaling` to be normally distributed and at least retains the original ordering. It removes all information about heteroskedasticity.

### Using GARCH

The quantile matching method above is not great but the calculation of `scaling` can be done in other ways too. If you were to fit a GARCH model on the return series and calculate the volatility for each period you could set

```
scaling <- 1 / volatility
```

and proceed as above.

To mimic Gaussian returns in your portfolio, you can scale your portfolio by `scaling` as well.

### Using factor models

The above methods work on time series data, if your interested in the returns on multiple stocks at one moment you can use a factor model. For example, the CAPM. If the CAPM was correct, returns would be given by: $$R_i = R_f + \beta_i(R_m - R_f) + \varepsilon_i$$ where $R_i$ denotes the return on asset $i$, $R_f$ the risk free rate, $\beta_i = \frac{\mathrm{Cov}(R_m, R_i)}{\mathrm{Var}(R_m)}$ the scaled covariance of the returns of $i$ with the market, $R_m$ the market return and $\varepsilon_i$ the return on $i$ not explained by the model. If the model would be well specified $\varepsilon_i$ is normally distributed with mean $0$ and standard deviation $\sigma_i$. Then $$R_i \sim N(R_f + \beta_i(R_m - R_f), \sigma_i).$$

Other factor models are the Fama & French 3 factor model or the Carhart four-factor model.

However, these factor models don't really get rid of fat tails as they don't capture all there is to know about returns.