Power and Bipower Variation with Stochastic Volatility and Jumps

TLDR

Detailed mathematical results are reported in Barndorff‑Nielsen and Shephard (2003a). The authors show that for a stochastic volatility process with infrequent jumps, the difference between realized variance and realized bipower variation estimates the jump component’s quadratic variation, and they provide various extensions and proofs of special cases. The study demonstrates that realized bipower variation, introduced here, is robust to rare jumps, can estimate integrated variance in stochastic volatility models as a model‑free alternative to realized variance, and is the first method capable of separating continuous and jump components of quadratic variation.

Abstract

This article shows that realized power variation and its extension, realized bipower variation, which we introduce here, are somewhat robust to rare jumps. We demonstrate that in special cases, realized bipower variation estimates integrated variance in stochastic volatility models, thus providing a model-free and consistent alternative to realized variance. Its robustness property means that if we have a stochastic volatility plus infrequent jumps process, then the difference between realized variance and realized bipower variation estimates the quadratic variation of the jump component. This seems to be the first method that can separate quadratic variation into its continuous and jump components. Various extensions are given, together with proofs of special cases of these results. Detailed mathematical results are reported in Barndorff-Nielsen and Shephard (2003a).