Good and bad volatility estimation for drift diffusion process

The logarithmic prices of financial assets are conventionally assumed to follow a drift-diffusion process. While the drift term is typically ignored in infill asymptotic theory and its applications, the presence of nonzero drift is an undeniable reality. Our finite-sample theory, along with extensive simulations, reveals the non-negligible impact of drift on the estimation precision of good and bad volatility. We also demonstrate that this poor estimation of good and bad volatility leads to significant bias in signed jump estimation. As a solution, we propose an alternative construction of good volatility, bad volatility, and signed jumps, which shows a marked improvement in estimation accuracy in the presence of non-negligible drift. When applying the modified estimators to forecast stock market volatility, we do not find evidence of an asymmetric impact of good and bad volatility, nor a significant role for signed jumps. We show that the asymmetric effects of good and bad volatility, as well as the role of signed jumps reported in existing literature, may be largely attributed to biases in their estimators caused by a nonzero drift.