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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?

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    $\begingroup$ Please make acronyms like PMCMC explicit or provide a link. Where did you get this volatility model from? Did you estimate an $x_0$ as well or do you assume $x_0 = 0$? You're assuming that the spread is always positive here, it this what you want? $\endgroup$ – SRKX Jun 18 '15 at 4:39
  • $\begingroup$ - This model is the standard stochastic volatility. You can find content on arxiv.org/pdf/1408.6980.pdf (p. 13). I assume $x_0 = 0$. The spread can be positive/negative since it's a linear combination of asset prices (The most important in this analysis is the stationarity of the spread). $\endgroup$ – Philippe Remy Jun 18 '15 at 10:10
  • $\begingroup$ What I think is: Because $p_t | x_t, p_{t-1} = p_{t-1} \exp( N(0, \beta^2 exp(x_t)))$ So with a simple Monte Carlo, I sample from this dist to construct the volatility associated to $p_t$ with regard to $x_t, p_{t-1}$. $\endgroup$ – Philippe Remy Jun 18 '15 at 11:11
  • $\begingroup$ With this model, you have $p_t = p_{t-1} \exp(y_t)$, and this means that if $p_t$ is the price of the spread (as I think you do), then it will never change sign. $\endgroup$ – SRKX Jun 19 '15 at 1:23
  • $\begingroup$ You're right! My model only permits positive values for the spread. So I need another formula to compute the returns. It should be more something like $y_t = |p_t / p_{t-1}| - 1$ $\endgroup$ – Philippe Remy Jun 19 '15 at 21:35
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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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  • $\begingroup$ Thanks for the help but I'm more keen on to model the volatility of $p_t$ instead of $y_t$. Do you know how I can proceed? $\endgroup$ – Philippe Remy Jun 18 '15 at 9:58
  • $\begingroup$ $y_t$ is the return, the different between consecutive price $p_t$. So you want to model the volatility of the price instead of return? If that's case, it's typically un-doable because financial price is non-stationary (not refers to heteroskedasticity, but the trending component in a price time series) and the volatility for non-stationary time-series is undefined. $\endgroup$ – Jianxun Li Jun 18 '15 at 12:12
  • $\begingroup$ If you are modelling spread (price difference between different assets at the same time), it has a chance that those two assets are co-integrated (and hence suitable for pair-trading). In that case, you can model the volatility of the spread either by this standard SV model or a regime-switching SV model. $\endgroup$ – Jianxun Li Jun 18 '15 at 12:14
  • $\begingroup$ Yes that's exactly what I'm doing. My spread $p_t$ is stationary. I use $p_t|x_t,p_{t-1} = p_{t-1} \exp(N(0, \beta^2 \exp(x_t)))$, because $y_t = \log(p_t) - \log(p_{t-1})$ and a simple Monte Carlo method to sample from this dist to have the hidden volatility associated to $p_t$. I can't input $p_t$ to a standard SV model, do I? $\endgroup$ – Philippe Remy Jun 18 '15 at 12:43
  • $\begingroup$ As a matter of fact, my goal is to model the volatility of the spread. $\endgroup$ – Philippe Remy Jun 18 '15 at 12:49

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