Fitting Stochastic Volatility to Returns and Matching Stylized Facts

Summary

The document asks how to estimate a stochastic volatility model from historical daily returns without using an implied volatility surface or computationally intensive MCMC. It specifies a latent log variance that follows an autoregressive process with Gaussian innovations, while returns conditional on that variance follow a Student t distribution. A proposed approach is joint maximum likelihood over the latent variance states and model parameters, potentially supplemented by moment matching between observed and simulated returns.

The intended fit should reproduce several features of financial returns: persistent, slowly decaying dependence in log variance; negative lag-one dependence in changes to log variance; and negative association between a return shock and the next period's absolute return. The author suggests approximating long memory with multiple exponential components and notes that leverage effects may require a richer model. The document presents modeling goals and a candidate likelihood setup, but does not report an estimation procedure, fitted parameters, empirical results, or a demonstrated assessment of whether the proposed method works. Its stated use is simulation of future price paths for risk analysis such as historical-simulation VaR.

Key ideas

• The proposed model combines latent autoregressive log variance with conditionally Student t returns.
• Joint maximum likelihood over latent states is suggested as a simpler alternative to MCMC.
• A suitable fit should capture volatility persistence, zigzagging variance changes, and negative shock asymmetry.
• Multiple exponential components may approximate long memory within a tractable model.
• The document outlines desired behavior but provides no fitted results or validated estimation method.

Tags

# Fit SV on historical data, simple and suboptimal but structurally correct


# Fit SV on historical data, simple and suboptimal but structurally correct












Is there a simple way to fit a Stochastic Volatility model directly to historical returns (not an IV surface), without heavy machinery like MCMC?

The fit can be suboptimal, but it should be structurally correct (details below).

Baseline SV model on daily log returns:

$$ h_{t+1} = \omega + \beta h_t + \sigma_h \eta_{t+1}, \qquad \eta_{t+1} \sim \mathcal N(0,1). $$

$$ r_{t+1} \mid h_{t+1} \sim t_\nu\!\left(\mu,\; e^{h_{t+1}/2}\right). $$

Joint Fit Approach. I am wondering whether a GARCH-like joint MLE fit (treating $h_t$ as latent states) can produce something reasonable? Possibly with additional moment matching constraint to match moments of real and simulated data.

A single-step conditional log-likelihood would be

$$ \ell(h_{t+1}, r_{t+1} \mid h_t) = \log \mathcal N\!\left( h_{t+1};\, \omega + \beta h_t,\, \sigma_h^2 \right) + \log t_\nu\!\left( r_{t+1};\, \mu,\, e^{h_{t+1}/2} \right). $$

What I mean by “structurally correct”

When comparing moments of real returns with those generated by the model, there should be no large single errors (e.g. under $L_2$ or preferably $L_∞$ norms), it's ok to have many small errors.

Stylised facts (moments) the model should reproduce:

- Long memory in $\log σ^2$: positive autocorrelation of $\log σ^2$ with polynomial decay (the actual model would be more complex, long memory would be approximated using two exponential components).

- Zigzag behaviour in $Δ \log σ^2$: negative lag-1 autocorrelation of $Δ \log σ^2$.

- Negative shock asymmetry: $\text{COR}[r_t, |r_{t+1}|] < 0$ (actual model will be more complex, with correlation for neg shock).

Use case

Simulating possible future price paths, like historical simulation of VaR.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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