Stochastic volatility filtering

Volatility you cannot see

Returns come in calm stretches and in storms, so the spread of a return changes from day to day, and nobody reports it. The stochastic volatility model treats the log of the variance, $h_t$, as a hidden state. It moves as a first-order autoregression around a long-run level $\mu$: a fraction $\varphi$ of yesterday's gap stays, and a fresh shock of size $\sigma$ arrives. The return is a bell curve with variance $e^{h_t}$. Using the log keeps the variance positive and makes shocks multiply it. Three numbers describe the model: the level $\mu$, the persistence $\varphi$ (near $1$ means volatility shocks last) and the volatility of volatility $\sigma$.

The hidden state sits inside an exponential, so the observation is a non-linear function of it and the Kalman filter has no exact answer. A bootstrap particle filter does. Start the particles from the long-run distribution of $h$. Each day, move every particle with the autoregression, weight it by the normal density of the return with variance $e^{h}$, and resample when the effective sample size is low. The filtered volatility is the square root of the weighted average of $e^{h}$, and multiplying by $\sqrt{252}$ turns a daily figure into an annual one. The average of the particle likelihoods also gives the log-likelihood of the parameters, which the next topic uses.

On the $5{,}404$ SPY returns since 2005, with $\mu=\ln(\overline{r^{2}})=-8.85$, $\varphi=0.98$, $\sigma=0.15$ and $1000$ particles, the filtered annual volatility averages $8.7\%$ in 2017 and $30.8\%$ in 2008, and peaks at $79.5\%$ on 15 October 2008. The log-likelihood is $17{,}752$, against $16{,}252$ for a constant volatility, a gain of about $1{,}500$. The effective sample size has a median of $966$ out of $1000$, but it falls to $3.3$ on 10 October 2018 ($-3.2\%$ after a calm spell, the filtered volatility jumping from $7.9\%$ to $18.9\%$). After calm, almost no particle has a high enough volatility to explain a shock.

• Hidden state is the log-variance: it keeps the variance positive and makes shocks multiply it.
• Weights favour the volatility that explains the return: a big return lifts the stormy particles and a small one lifts the calm ones.
• Watch the ESS after calm: a big surprise can leave very few useful particles.

Lab · what's graded

• The long-run spread15 pts
• Annual volatility of a particle10 pts
• The predicted mean15 pts
• A normal density10 pts
• A weight ratio15 pts
• Why the log-variance10 pts
• Match the vocabulary10 pts
• Why not just use the return15 pts