Simulating Stock Returns with GARCH and a Student-t Copula

Summary

The document describes a proposed scenario-generation pipeline for stock returns. It fits a univariate GARCH(1,1) model with an ARX mean and Student-t residuals to each stock, transforms standardized residuals to uniform variables, fits a multivariate Student-t copula, then simulates copula draws and maps them back through each fitted marginal distribution. The simulated shocks are combined with the GARCH conditional means and volatilities to reconstruct returns.

The motivating problem is that the resulting series contains extremely large outliers, prompting the question of whether to remove them. The excerpt provides no answer or diagnostic evidence, so it does not establish whether the cause is a fitting or simulation error, tail behavior, or implementation issue. It documents the setup and the concern, but gives no validated remedy or results supporting observation deletion.

Key ideas

• The proposed pipeline fits separate GARCH models before estimating dependence with a Student-t copula.
• Probability integral transforms place marginal residuals on a uniform scale for copula fitting.
• Simulated copula draws are mapped back through the fitted Student-t marginals to reconstruct returns.
• Extreme simulated returns are reported, but the document gives no diagnosis or recommendation about removing them.

Tags

# Simulating stock returns GARCH-copula approach


# Simulating stock returns GARCH-copula approach












I am currently in need of simulating stock returns from 2025 until 2100 for scenario analysis purpose. I used a GARCH-copula approach : mean = ARX for GARCH, student t residuals and student t copula. I followed this procedure :

- Fit a univariate GARCH(1,1) model with student-t residuals to each stock $S_i$

- Find parameters of the distribution of residuals for each stock

- Apply the Probability integral transform to residuals to make the residuals follow the $Unif[0,1]$ law.

- Fit a multivariate student-t copula to the uniform residuals of all stocks.

- Simulate D=75.365 occurences of the copula

- Convert back to student by applying t.ppf to each column of the copula simulations with associated parameters and obtain eps_t, t=1,...,nbr stocks

- Reconstruct using mu and sigma from GARCH - $r_t = mu + phi * r_{t-1} + sigma_t * eps_t$

The problem is that at the end, I obtain some outlier returns that are of order $10^4$ and even bigger for example. Is there a way to prevent this? Or should I just drop those observations?

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.