Volatility under SSM

Volatility under SSM#

We talk about a Gaussian state-space model for volatility.

The model is given by

\[ h_t = \mu + \phi h_{t-1} + \eta_t, \quad \eta_t \sim \mathcal{N}(0, \sigma_\eta^2) \]

Where $h_t$ is log-volatiltiy, $\mu$ is long-term mean, $\phi \in [-1,1]$ imposes exponential decay, and $\eta$ is Gaussian noise.

We note that this is a mean-reverting AR(1) process, where $h_t$ can also be seen as an exponentially weighted moving average (EWMA).

We also assume this is latent, and we only observe the log returns conditional on volatility:

\[ r_t = \epsilon_t e^{h_t / 2}, \quad \epsilon_t \sim \mathcal{N}(0,1) \]

or equivalently $$ r_t \sim \mathcal{N}(0,e^{h_t}) $$

Where return has zero mean and variance given by volatility

Consider this re-parametrization using more familiar notation: $$ x_0 \sim \mathcal{N} \left( x_0; \mu, \frac{\sigma_v^2}{1 - \phi^2} \right), $$

\[ x_{t+1} | x_t \sim \mathcal{N} \left( x_{t+1}; \mu + \phi (x_t - \mu), \sigma_v^2 \right), \]

\[ y_t | x_t \sim \mathcal{N} \left( y_t; 0, \exp(x_t) \right), \]

Where $x_t$ is log-volatility and $y_t$ is observed log return

We observed $y_{1:T}$ and wish to infer $x_{0:T}$

Notice volatility clustering can be captured in the model by setting $|\phi|$ close to 1

Implemented according to this paper