Stochastic Volatility and Particle Filtering

Worked Solution

How to Think About It: This is a Hidden Markov Model where the latent state $h_t$ (log-volatility) drives the observable returns. You never see $h_t$ directly -- you infer it from the noisy signal $r_t$. Kalman filtering would work if both equations were linear-Gaussian, but the observation equation $r_t = e^{h_t/2} \varepsilon_t$ is nonlinear (the variance is $e^{h_t}$, which is a nonlinear function of $h_t$). This is why we need a particle filter -- it approximates the posterior $p(h_t \mid r_{1:t})$ with a weighted cloud of samples rather than a Gaussian.

Key Insight: The log-likelihood factors as $\log p(r_{1:T}) = \sum_{t=1}^{T} \log p(r_t \mid r_{1:t-1})$, and the particle filter naturally estimates each predictive density term $p(r_t \mid r_{1:t-1})$ as a byproduct of the resampling step.

Part 1 -- State-Space Form and Bootstrap Particle Filter:

State-space form: - Observation equation: $r_t \mid h_t \sim N(0, e^{h_t})$ - State transition: $h_t \mid h_{t-1} \sim N(\mu + \rho(h_{t-1} - \mu),\ \sigma_\eta^2)$ - Initial state: $h_0 \sim N(\mu,\ \sigma_\eta^2 / (1 - \rho^2))$ (stationary distribution)

Bootstrap particle filter with $M$ particles $\{h_t^{(i)}\}_{i=1}^{M}$:

1. Initialize ($t = 0$): Draw $h_0^{(i)} \overset{\text{iid}}{\sim} N(\mu,\ \sigma_\eta^2 / (1-\rho^2))$ for $i = 1, \ldots, M$.

2. Propagate (predict step): For each particle, draw from the transition: $\tilde{h}_t^{(i)} \sim N\!\left(\mu + \rho(h_{t-1}^{(i)} - \mu),\ \sigma_\eta^2\right)$

3. Weight (update step): Compute unnormalized weights using the observation likelihood: $w_t^{(i)} \propto p(r_t \mid \tilde{h}_t^{(i)}) = \frac{1}{\sqrt{2\pi e^{\tilde{h}_t^{(i)}}}} \exp\!\left(-\frac{r_t^2}{2 e^{\tilde{h}_t^{(i)}}}\right)$ Normalize: $\bar{w}_t^{(i)} = w_t^{(i)} / \sum_j w_t^{(j)}$.

1. Resample: Draw $M$ indices with replacement from $\{1,\ldots,M\}$ with probabilities $\{\bar{w}_t^{(i)}\}$. Set $h_t^{(i)}$ to the selected particles. (Systematic resampling is preferred for lower variance.)

1. Repeat from step 2 at time $t+1$.

The particle approximation to the filtering distribution is $p(h_t \mid r_{1:t}) \approx \frac{1}{M} \sum_{i=1}^{M} \delta(h_t - h_t^{(i)})$.

Part 2 -- One-Step-Ahead Variance Forecast:

The forecast $E[\sigma_{t+1}^2 \mid r_{1:t}] = E[e^{h_{t+1}} \mid r_{1:t}]$. Using the law of total expectation and the transition: $E[e^{h_{t+1}} \mid r_{1:t}] = E\!\left[E[e^{h_{t+1}} \mid h_t] \,\middle|\, r_{1:t}\right]$

Since $h_{t+1} \mid h_t \sim N(\mu + \rho(h_t - \mu),\ \sigma_\eta^2)$, the inner expectation is a log-normal moment: $E[e^{h_{t+1}} \mid h_t] = \exp\!\left(\mu + \rho(h_t - \mu) + \frac{\sigma_\eta^2}{2}\right)$

Plugging in the particle approximation: $E[\sigma_{t+1}^2 \mid r_{1:t}] \approx \frac{1}{M} \sum_{i=1}^{M} \exp\!\left(\mu + \rho(h_t^{(i)} - \mu) + \frac{\sigma_\eta^2}{2}\right)$

Part 3 -- Log-Likelihood via Particle Filtering:

The log-likelihood decomposes as: $\log p(r_{1:T}; \theta) = \sum_{t=1}^{T} \log p(r_t \mid r_{1:t-1}; \theta)$

Each predictive factor is estimated by the average weight before normalization at step $t$: $p(r_t \mid r_{1:t-1}; \theta) \approx \frac{1}{M} \sum_{i=1}^{M} w_t^{(i)}$

where $w_t^{(i)} = p(r_t \mid \tilde{h}_t^{(i)})$ are the unnormalized weights. The particle log-likelihood estimate is: $\widehat{\log p}(r_{1:T}; \theta) = \sum_{t=1}^{T} \log\!\left(\frac{1}{M} \sum_{i=1}^{M} w_t^{(i)}\right)$

For MLE, treat this as an objective function in $\theta = (\mu, \rho, \sigma_\eta^2)$ and maximize numerically. Because the estimate is noisy (different across runs), use iterated filtering (IF2) or particle MCMC rather than vanilla gradient methods. With $M = 1000$--$5000$ particles, the log-likelihood estimate is reasonably smooth and gradient-free optimizers (Nelder-Mead, differential evolution) work well.

Answer: - Particle filter: propagate from prior, weight by observation likelihood $N(0, e^{h_t})$, resample. - Forecast: $E[\sigma_{t+1}^2 \mid r_{1:t}] \approx \frac{1}{M} \sum_i \exp(\mu + \rho(h_t^{(i)} - \mu) + \sigma_\eta^2/2)$. - Log-likelihood: sum of log average weights; maximize over $\theta$ using derivative-free optimization.