  Volatility clustering is a common phenomenon in financial time series.
Typically, linear models can be used to describe the temporal autocorrelation
of the (logarithmic) variance of returns. Considering the difficulty in
estimating this model, we construct a Dynamic Bayesian Network, which utilizes
the conjugate prior relation of normal-gamma and gamma-gamma, so that its
posterior form locally remains unchanged at each node. This makes it possible
to find approximate solutions using variational methods quickly. Furthermore,
we ensure that the volatility expressed by the model is an independent
incremental process after inserting dummy gamma nodes between adjacent time
steps. We have found that this model has two advantages: 1) It can be proved
that it can express heavier tails than Gaussians, i.e., have positive excess
kurtosis, compared to popular linear models. 2) If the variational
inference(VI) is used for state estimation, it runs much faster than Monte
Carlo(MC) methods since the calculation of the posterior uses only basic
arithmetic operations. And its convergence process is deterministic.
  We tested the model, named Gam-Chain, using recent Crypto, Nasdaq, and Forex
records of varying resolutions. The results show that: 1) In the same case of
using MC, this model can achieve comparable state estimation results with the
regular lognormal chain. 2) In the case of only using VI, this model can obtain
accuracy that are slightly worse than MC, but still acceptable in practice; 3)
Only using VI, the running time of Gam-Chain, in general case, can be reduced
to below 5% of that based on the lognormal chain via MC.
