Estimation of the Continuous and Discontinuous Leverage Effects

2 For example, extensive literatures in finance and econometrics are based on the model proposed by Heston (Citation1993), including (Yu Citation2005; Vetter Citation2012; Aït-Sahalia, Fan, and Li Citation2013).

3 This problem has also been studied by, for example, Fan and Wang (Citation2008) and Alvarez et al. (Citation2012). Moreover, Jacod and Todorov (Citation2010) and Li and Xiu (Citation2016) included the estimation of spot volatility as an intermediate step to their main goals. Just like the last two cited papers, the main target of this article is not the estimation of the spot volatility process either, and nontrivial results follow right after we obtain the estimator of the spot volatility process.

4 Different from CLE and DLE estimation, without correction, a bias is still present in the case of leverage parameter estimation, because the quadratic variation of estimated spot volatility involves squared estimated volatility changes.

5 More specifically, we have $\left(\delta 1_{\left\{\right|\delta |\le \kappa \}}\right)★{(\mu -\nu )}_t={\int }_0^t{\int }_{\mathbb{R}}\left(\delta (\omega ,s,x)1_{\left\{\right|\delta (\omega ,s,x)|\le \kappa \}}\right)(\mu -\nu )(ds,dx)$ and $\left(\delta 1_{\left\{\right|\delta |>\kappa \}}\right)★{\mu }_t={\int }_0^t{\int }_{\mathbb{R}}\left(\delta (\omega ,s,x)1_{\left\{\right|\delta (\omega ,s,x)|>\kappa \}}\right)\mu (ds,dx)$.

6 In the definition of the co-jump leverage in Bandi and Renò (Citation2012), the jump intensity is further related to the level of the volatility (or variance), and the random jump sizes (in return and volatility series) are replaced by their correlation. As a result, the first source of randomness is restricted, although such an assumption embeds an interesting driving force in the dynamics of the jump intensity process, while the second source of randomness is eliminated.

7 In Jacod and Todorov (Citation2010), the authors studied a very general limit functional U(F)_T (with minor changes in notations in what follows) given by (3.4) $$\begin{array}{c}\hfill U{\left(F\right)}_T=\sum _{s\le T}F(\Delta X_s,{\sigma }_{s-}^2,{\sigma }_s^2)1_{\{\Delta X_s\ne 0\}},\end{array}$$(3.4) together with its estimator (3.5) $$\begin{array}{c}\hfill U{(F,k_n)}_T=\sum _{i=k_n+1}^{\lfloor T/{\Delta }_n\rfloor -k_n}F\left({\Delta }_i^{n}X,{\widehat{\sigma }}_{i-}^2,{\widehat{\sigma }}_{i+}^2\right)1_{\left\{\right|{\Delta }_i^{n}X|>{\alpha }_n\}}.\end{array}$$(3.5) Here F is a function on $\mathbb{R}\times {\mathbb{R}}_{{+}^{*}}\times {\mathbb{R}}_{{+}^{*}}$ where ${\mathbb{R}}_{{+}^{*}}=(0,\infty )$. Observe that with the specification F(x, y, z) = x(z − y), (Equation3.4(3.4) $$\begin{array}{c}\hfill U{\left(F\right)}_T=\sum _{s\le T}F(\Delta X_s,{\sigma }_{s-}^2,{\sigma }_s^2)1_{\{\Delta X_s\ne 0\}},\end{array}$$(3.4) ) and (Equation3.5(3.5) $$\begin{array}{c}\hfill U{(F,k_n)}_T=\sum _{i=k_n+1}^{\lfloor T/{\Delta }_n\rfloor -k_n}F\left({\Delta }_i^{n}X,{\widehat{\sigma }}_{i-}^2,{\widehat{\sigma }}_{i+}^2\right)1_{\left\{\right|{\Delta }_i^{n}X|>{\alpha }_n\}}.\end{array}$$(3.5) ) would be equivalent to (Equation2.5(2.5) $$\begin{array}{c}\hfill [X,{\sigma }^2{]}_T^D:=\sum _{t\le T}\Delta X_t\Delta {\sigma }_t^2.\end{array}$$(2.5) ) and (Equation3.6(3.6) $$\begin{array}{c}\hfill [\widehat{X,{\sigma }^2}{]}_T^D=\sum _{i=k_n+1}^{\lfloor T/{\Delta }_n\rfloor -k_n}{\Delta }_i^n{X}^{{\alpha }_n}({\widehat{\sigma }}_{i+}^2-{\widehat{\sigma }}_{i-}^2).\end{array}$$(3.6) ) (the discontinuous leverage effect and its estimator), respectively. As we will see, however, the specific functional form of F will allow us to derive central limit results that apply under much broader conditions regarding jump stochasticity than those applying to a generic F.

8 Here, $Z_T^n\stackrel{u.c.p.}{\to }{Z}_T$ means that the sequence of stochastic processes Zⁿ_T converges in probability, locally uniformly in time, to a limit Z_T, that is, ${sup}_{s\le T}\mid Z_s^n-Z_s\mid \stackrel{\mathbb{P}}{⟶}0$ for all finite T.

9 Refer to the decomposition of X in Equation (A.1) and note that the conditional expectation of (Δⁿ_jX′′)σ_tⁿj²Δ_n is of the order Δ²_n, the same as (Δⁿ_jX′)σ_tⁿj²Δ_n.

10 But in Bandi and Renò (Citation2012), volatility jumps will slow down the convergence rate.

11 A possible explanation is that the Dow Jones Index is constructed from very liquid stocks. In fact, the realized variation first stays roughly the same as we increase the sampling frequency from 30-min to about 3-min, and then decreases (rather than increases) as we further increase the frequency to 5 sec. This is consistent with the finding that 5-sec returns are positively correlated, which might be explained by self- and/or mutual excitation in jumps (see, e.g., Aït-Sahalia, Cacho-Diaz, and Laeven Citation2015; Boswijk, Laeven, and Yang Citation2015). For instance, it could be the case that occurred jumps tend to excite other jumps with the same sign in the short run. If such scenario happens frequently, then the sample autocorrelations can be positive. Yet we don’t expect such same-sign-jump-exciting effect to last long, perhaps only a few minutes.

12 See also Fan and Fan (Citation2011) for an improved version of the test.