基于链式Gamma分布的随机游走波动率模型
Modeling Randomly Walking Volatility with Chained Gamma Distributions
摘要：
波动率聚集是金融时间序列中的一种普遍现象。通常，线性模型可以用来描述收益率（对数）方差的时间自相关性。考虑到该模型估计的困难性，我们构造了一个动态贝叶斯网络，它利用了正态-gamma和gamma-gamma的共轭先验关系，使得其后验形式在每个节点上局部保持不变。这使得使用变分方法快速找到近似解成为可能。在相邻时间步之间插入虚拟Gamma节点后，保证了模型所表达的波动率是一个独立的增量过程。我们发现这个模型有两个优点：1）可以证明它可以表达比高斯模型更重的尾部，即，与流行的线性模型相比，具有正的超额峰度。2)如果变分推理（VI）用于状态估计，它比蒙特卡罗（MC）方法运行得快得多，因为后验的计算仅使用基本的算术运算。其收敛过程是确定性的。我们测试了这个名为Gam-Chain的模型，使用了不同分辨率的加密货币、纳斯达克和外汇交易记录。结果表明：1）在相同的MC条件下，该模型可以获得与正则对数正态链相当的状态估计结果。2)在仅使用VI的情况下，该模型的精度略低于MC，但在实际应用中仍然可以接受; 3）仅使用VI，Gam-Chain的运行时间在一般情况下可以减少到基于MC的对数正态链的运行时间的5%以下。
英文摘要：
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.
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