Power and Bipower Variation with Stochastic Volatility and Jumps

Abstract

This paper shows that realised power variation and its extension called realised bipower variation that we introduce here is somewhat robust to rare jumps. We demonstrate that in special cases realised bipower variation estimate integrated variance in stochastic volatility models, thus providing a model free and consistent alternative to realised variance. Its robustness property means that if we have a stochastic volatility plus infrequent jumps process then the difference between realised variance and realised bipower variation estimates the quadratic variation of the jump component. This seems to be the first method which 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 will be reported elsewhere.

Good and Bad Variance Premia and Expected Returns

“…Following Barndorff-Nielsen and Shephard (2004), realized variance, rv t , is defined as the sum of squared high-frequency returns, r t,i , within each period:…”

A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales

“…The bipower variation process of order (r, s) for Y , denoted by V (Y ; r, s) t , is the limit in probability, if it exists for all t ≥ 0, of V (Y ; r, s) n t . It has been introduced in [4] and [5], where it is shown that the bipower variation processes exist for all nonnegative indices r, s as soon as Y is a continuous semimartingale of "Itô type" with smooth enough coefficients. These papers also contain a version of the associated CLT under somewhat restrictive assumptions and when r = s = 1.…”

Do Co-Jumps Impact Correlations in Currency Markets?

“…We use the following benchmark estimators: the realized covariance (Eq.5), the bipower realized covariance of Barndorff-Nielsen and Shephard (2004b), the two-scale realized covariance of Zhang (2011), the multivariate realized kernel of BarndorffNielsen et al (2011), and our jump wavelet covariance estimator (Eq. 12).…”

“…With increased availability of high-frequency intraday data, the literature has shifted from parametric conditional covariance estimation toward modelfree measurement. This paradigm shift from treating covariances as latent towards directly modeling ex-post covariance measures constructed from intraday data (Andersen et al, 2003;Barndorff-Nielsen and Shephard, 2004b) has spurred additional interest. Although the theory is appealing and intuitive, it assumes that the observed high-frequency data represent the underlying process.…”