Using GARCH Residuals and Copulas for Portfolio Expected Shortfall
Summary
The document asks how to combine univariate GARCH volatility models with a copula to estimate expected shortfall for a portfolio of three stocks. It considers whether to obtain conditional volatility by filtering observed returns through each fitted GARCH recursion or by simulating a separate return path, then scaling the actual returns into residuals. It also asks how to transform residuals to uniform observations when marginal innovations differ, and how to fit a t-copula to those observations.
The proposed workflow uses each stock’s fitted marginal distribution to transform standardized residuals through its CDF, fits copula dependence from the resulting uniforms, and simulates joint outcomes for portfolio risk analysis. The text supplies no answer, data, or empirical comparison of the alternatives. It leaves unresolved details such as volatility initialization, parameter estimation, and the distinction between copula simulation and transforming simulated uniforms back through each marginal model.
Key ideas
- Conditional volatility for residual standardization should be filtered from observed returns using the fitted GARCH recursion.
- Each marginal innovation distribution determines the CDF used to map standardized residuals to uniform observations.
- A t-copula models dependence after marginal distributions have been transformed to uniforms.
- Portfolio expected-shortfall simulation requires converting dependent uniform draws back through the fitted marginal distributions.
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Full text
# Fitting a Copula with GARCH volatility to stock returns
# Fitting a Copula with GARCH volatility to stock returns
I have the log-returns $r_{n,t}$ for 3 stocks, $n=1,2,3$, and $t=1,..,T=365$ days, and I want to model the expected shortfall given arbitrary positions on those stocks.
I calibrate the GARCH model with student-t innovations using MLE, which gives me 3 sets of $\omega, \alpha, \beta, \nu$ (assuming $\mu=0$ in the returns). I then use my empirical data, $r_{n,t}$ and divide it by the conditional volatility, $\sigma_{n,t}$ to get the residuals, $\epsilon_{n,t}$.
- Where do I get the conditional volatility, $\sigma_{n,t}$ from?
Method 1: Run a simulation for 365 days (+ some burn for the $\sigma_0$ guess) using my calibrated parameters to get a time-series of $r^{sim}_{n,t}$ and $\sigma^{sim}_{n,t}$using: $$ r_{sim,t} \sim t_{\nu}(0,\sigma^2_t)$$ $$ \sigma_{sim,t}^2 = \omega + \alpha r_{sim,t-1}^2 + \beta \sigma_{sim,t-1}^2$$
for each stock, then divide the resultant $\sigma^{sim}_{n,t}$ by the real stock data to calculate $\epsilon_{n,t} =\frac{r^{real}_{n,t}}{\sigma^{sim}_{n,t}} $?
Method 2: input the real stock data into $\sigma_{n,t}^2 = \omega_n + \alpha_n r_{n,t}^2+ \beta_n \sigma_{n,t}^2$, then retrieve $\epsilon_{n,t} =\frac{r^{real}_{n,t}}{\sigma_{n,t}} $
I am not sure which method is correct.
- If I wanted to model stocks 1,2 as normal-innovations and stock 3 as t-innovations, when inserting the residuals, $\epsilon_t=\frac{r_t}{\sigma_t}$ into the CDF to get the uniforms, would I need to use the standard normal CDF and the standard student-t CDF? i.e. $[u_{1,t},u_{2,t},u_{3,t}]= [\Phi_{normal}(\epsilon_{1,t}),\Phi_{normal}(\epsilon_{2,t}), \Phi_t(\epsilon_{3,t})]$ for each time-step. Or am I supposed to only use 1?
- If I then wanted to fit a t-copula, how do I actually calibrate it?
From what I understand, for the case where we use t-innovations for all stocks, we would use the $u_{n,t}$ observations in question 2, and then we fit $\nu$ and correlation matrix $P$ of the copula using the PDF:
$$ c_{\nu, P}^t(u_{1,t},u_{2,t},u_{3,t})=\frac{f_{\nu, P}\left(t_\nu^{-1}\left(u_{1,t}\right), t_\nu^{-1}\left(u_{2,t}\right), t_\nu^{-1}\left(u_{3,t}\right)\right)}{\prod_{i=1}^3 f_\nu\left(t_\nu^{-1}\left(u_{i,t}\right)\right)} $$
But taking the inversion just gives us the residuals, what's the point in taking the inverse? How would this be different if we used 2 normal innovations and 1 student-t?
- Then to get the correlated observations using a monte-carlo simulation for expected-shortfall analysis, am I supposed to draw $r_{n,t} \sim c_{\nu,P}$ (the calibrated multivariate-t copula)?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.