  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.
