We model time series of VIX (monthly average) and monthly stock index returns. We use log-Heston model: logarithm of VIX is modeled as an autoregression of order 1. Our main insight is that normalizing monthly stock index returns (dividing them by VIX) makes them much closer to independent identically distributed Gaussian. The resulting model is mean-reverting, and the innovations are non-Gaussian. The combined stochastic volatility model fits well, and captures Pareto-like tails of real-world stock market returns. This works for small and large stock indices, for both price and total returns.

翻译：本文对VIX（月度平均值）与月度股票指数收益率的时间序列进行建模。我们采用对数Heston模型：将VIX的对数建模为一阶自回归过程。我们的核心发现是：将月度股票指数收益率除以VIX进行标准化处理后，其分布更接近独立同分布的高斯分布。所得模型具有均值回归特性，且其扰动项服从非高斯分布。该组合随机波动率模型拟合效果良好，能够捕捉真实股票市场收益率中类似帕累托分布的厚尾特征。该模型适用于不同规模的股票指数，且对价格收益率与总收益率均具有适用性。