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)?
