Volatility and intensity
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When studying models and estimators in the setting of high-frequency data, simplifying assumptions are typically imposed on the relation between the observation times and the observable process. In this paper we study a certain form of endogeneity of the observation times, namely that they depend on non-observable spot processes. The prototypical example is that of a stochastic volatility model observed at times which are in part determined by the stochastic volatility process. We introduce an estimator of the quadratic covariation between two spot process semimartingales, and apply this estimator to the problem of assessing the relation between the spot volatility and the intensity process governing the observations times. Consistency of this estimator is proved, and its convergence rate is derived. In an empirical study of the Apple stock over 21 trading days we find indications of a correlation between the spot volatility and the observation times.
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