We propose a stochastic volatility model for time series of curves. It is motivated by dynamics of intraday price curves that exhibit both between days dependence and intraday price evolution. The curves are suitably normalized to stationary in a function space and are functional analogs of point-to-point daily returns. The between curves dependence is modeled by a latent autoregression. The within curves behavior is modeled by a diffusion process. We establish the properties of the model and propose several approaches to its estimation. These approaches are justified by asymptotic arguments that involve an interplay between between the latent autoregression and the intraday diffusions. The asymptotic framework combines the increasing number of daily curves and the refinement of the discrete grid on which each daily curve is observed. Consistency rates for the estimators of the intraday volatility curves are derived as well as the asymptotic normality of the estimators of the latent autoregression. The estimation approaches are further explored and compared by an application to intraday price curves of over seven thousand U.S. stocks and an informative simulation study.

翻译：本文提出了一种针对曲线时间序列的随机波动率模型。该模型受日内价格曲线动态过程的启发，这类曲线既表现出日间的依赖关系，又包含日内的价格演化特征。曲线经过适当标准化后在函数空间中具有平稳性，是点对点日收益率的函数型类比。曲线间的依赖关系通过隐自回归过程建模，而曲线内的行为则由扩散过程刻画。我们建立了该模型的性质，并提出了若干估计方法。这些方法通过涉及隐自回归与日内扩散之间交互作用的渐近论证得到验证。渐近框架融合了日内曲线数量的递增与每日曲线观测离散网格的精细化。我们推导出日内波动率曲线估计量的一致性收敛速率，以及隐自回归估计量的渐近正态性。通过对七千余只美国股票的日内价格曲线进行实证应用，并结合具有启发性的模拟研究，进一步探讨并比较了上述估计方法的性能。