Estimating Stochastic Volatility from Posterior Latent States

Summary

The document explains how to interpret latent states estimated in a standard stochastic-volatility model for log returns. In the stated model, the conditional standard deviation of returns at time t is beta multiplied by the exponential of half the latent log-volatility state. Thus, after fitting the model with particle Markov chain Monte Carlo, the latent state can be translated into an estimate of return volatility.

Because the latent state is inferred rather than observed, the suggested summary is to evaluate volatility across posterior draws and average those values for each time point. This produces a posterior estimate of the time-varying standard deviation, rather than a direct measure of volatility in the asset’s price level. The source’s displayed averaging expression omits the half exponent present in its model definition, so care is needed to apply the stated parameterization consistently. It gives no diagnostics or empirical example.

Key ideas

• The conditional standard deviation of returns is determined by beta and the latent log-volatility state.
• Posterior draws of the latent state can be transformed into volatility estimates at each time.
• Averaging transformed posterior draws summarizes uncertainty in the inferred volatility.
• The model describes return volatility, not volatility of the price level itself.

Tags

# How to use calibrated Standard Stochastic Volatility?


# How to use calibrated Standard Stochastic Volatility?












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?

## Answer by Jianxun Li (score 1)

https://quant.stackexchange.com/a/18429

The volatility of your asset $y_t$ is simply its time varying standard deviation, given by $\beta \exp(x_t/2)$. Once you've got the estimates for latent factor $x_t$ from converged MCMC chain, calculate the expected value for volatility at time $t$ using $$ \hat{v_t} = \mathbb{E}[\beta \exp(x_t/2)] = \frac{1}{R}\sum_{r=1}^R \beta \exp(x_t^{(r)}) $$ where $R$ is the total number of MCMC chains you've got and $x_t^{(r)}$ is the value of $x_t$ in $r$-th chain.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.