We introduce a new microstructure noise index for financial data. This index, the computation of which is based on the p-variations of the considered asset or rate at different time scales, can be interpreted in terms of Besov smoothness spaces. We study the behavior of our new index using empirical data. It gives rise to phenomena that a classical signature plot is unable to detect. In particular, with our data set, it enables us to separate the sampling frequencies into three zones: no microstructure noise for low frequencies, increasing microstructure noise from low to high frequencies, and some kind of additional regularity on the finest scales. We then investigate the index from a theoretical point of view, under various contexts of microstructure noise, trying to reproduce the facts observed on the data. We show that this can be partially done using models involving additive correlated errors or rounding error. Accurate reproduction seems to require either both kinds of error together or some unusual form of rounding error.
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