论文标题
用链式伽马分布对随机行走的波动进行建模
Modeling Randomly Walking Volatility with Chained Gamma Distributions
论文作者
论文摘要
波动性聚类是财务时间序列中的常见现象。通常，线性模型可用于描述返回（对数）差异的时间自相关。考虑到估计该模型的困难，我们构建了一个动态的贝叶斯网络，该网络利用了正常γ和伽马 - 伽马的共轭物关系，以使其后形式在每个节点上保持不变。这使得可以快速使用变分方法找到近似解决方案。此外，我们确保模型表达的波动率是在相邻时间步长之间插入虚拟伽马节点后的独立增量过程。我们发现该模型具有两个优点：1）可以证明，与流行的线性模型相比，它比高斯（即具有积极的多余峰度）可以表达较重的尾巴。 2）如果使用变异推理（VI）进行状态估计，则其运行速度比Monte Carlo（MC）方法快得多，因为后验的计算仅使用基本的算术操作。它的收敛过程是确定性的。 我们使用最近的加密货币，纳斯达克和外汇记录来测试该模型，名为Gam-Chain。结果表明：1）在使用MC的同一情况下，该模型可以通过常规对数正态链实现可比的状态估计结果。 2）仅使用VI，该模型可以获得比MC稍差的准确性，但在实践中仍然可以接受； 3）仅使用VI，通常，基于通过MC的对数正态链，可以将游戏链的运行时间降低至5％以下。
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.
