I'm considering the standard stochastic volatility model:
$$x_t = \rho x_{t-1} + \sigma \epsilon_x$$ $$y_t = \beta \exp\left[ \frac{x_t}{2} \right] \epsilon_y$$
where $y_t$ is the log-returns and $x_t$ the log-vol associated to $y_t$.
I used PMCMC to estimate $\rho, \sigma, \beta$.
My question is:
My target is to model the volatility of an asset (equity spread) $(p_t)_t$ based on this model. $y_t$ is calculated this way:
$$ y_t = \log(p_t) - \log(p_{t-1}) $$
Now that I estimated the latent variables, I get $x_t$, I don't know how I can get back the volatility of the spread $p_t$.
Can you help me out?