Double Conditional Smoothing of High-Frequency Volatility Surface Under a Spatial Model

Abstract

This article investigates a spatial model to analyze high-frequency returns in a nonparametric way. This model allows us to study the slow change of the volatility over a long period of time as well as the daily volatility patterns at the same time. A double conditional kernel regression is introduced to estimate the mean as well as the volatility surface. The idea is to smooth the data over the time of day on a given day in a first step. Those results are then smoothed over all observation days in a second step. It is shown that our proposal is equivalent to a common two-dimensional kernel regression. However, it runs much quicker than the traditional approach. Moreover, the first conditional smoothing also provides useful intermediate results. This idea works both for ultra-high frequency data and for adapted equidistant high frequency data. Asymptotic results for the proposed estimators in the latter case are obtained under suitable conditions. Selected examples show that the proposal works very well in practice. The impact of the 2008 financial crisis on the volatility surface is also discussed briefly.