Volatility estimation and jump detection for drift–diffusion processes

Abstract

Logarithms of prices of financial assets are conventionally assumed to follow driftdiffusion processes. While the drift term is typically ignored in the infill asymptotic theory and applications, the presence of nonzero drifts is an undeniable fact. The finite sample theory and extensive simulations provided in this paper reveal that the drift component has a nonnegligible impact on the estimation accuracy of volatility and leads to a dramatic power loss of a class of jump identification procedures. We propose an alternative construction of volatility estimators and jump tests and observe significant improvement of both in the presence of nonnegligible drift. As an illustration, we apply the new volatility estimators and jump tests, along with their original versions, to 21 years of 5-minute log-returns of the NASDAQ stock price index.

Unit Root Test With High-Frequency Data

“…However, the LM jump test suffers from a significant downward size distortion and has low power in finite samples when the drift coefficient µ t is large in size, as shown by Laurent and Shi (2020). An example given by Laurent and Shi (2020) is the drift-diffusion process (3) with θ and hence the drift coefficient µ t = θ (y t − µ) being nonzero. 10 They propose a simple modification to the LM test and show a dramatic improvement in test performance.…”

“…As shown in our empirical application, deviations from the random walk (i.e., nonzero θ ) are not rare events. It is, therefore, important to account for such a feature and rely on the jump identification procedure of Laurent and Shi (2020), which is less sensitive to µ t .…”

“…Let r i = y i − y (i−1) denote the log return at time i , mi be the median of the past M log returns (prior to and including the current observation), and r * i = r i − mi be the centered log return. The test statistic of Laurent and Shi (2020), denoted by U i , is constructed from the centered log returns (instead of the raw return r i as in LM) such that…”

“…As in Andersen, Bollerslev, and Dobrev (2007b), Boudt, Croux, andLaurent (2011), andShi (2020), we also take the intraday periodicity in the volatility into consideration. The jump test statistic is…”

“…Moreover, deviations from the random walk imply the presence of a nonzero drift in a linear drift-diffusion (e.g., Ornstein-Uhlenbeck [OU]) process. Laurent and Shi (2020) show that the presence of a nonzero drift results in the overestimation of the integrated variance using various realized volatility estimators (including jump robust estimators) and a power loss for jump detection procedures. As a remedy, they suggest using centered returns for the calculation of integrated volatilities and the construction of the jump test statistics.…”