3.3 Volatility prediction and microstructure noise
3.3.3 Dependent microstructure noise
So far we assumed that microstructure noise was i.i.d. While this assumption is one of mathematical convenience, it is not realistic as for example noted in recent work by A¨ıt-Sahalia, Mykland and Zhang (2005a) and Hansen and Lunde (2006) among others. It is the purpose of this subsection to digress on the case of dependent microstructure noise. For the purpose of presentation we confine our attention to the case of “plain vanilla” realized volatility.
We need to specify a new noise process. To do so, we adopt the model proposed by A¨ıt-Sahalia, Mykland and Zhang (2005a), namely we assume that the market mi- crostructure noise ηt process is a mixture of a white noise componentξt and an AR(1) componentφt. We use this dependence structure to derive the behavior of the market microstructure noise variance. In particular, we assume that:
Assumption 3.3.2 Consider the process η in equation (3.1) which is replaced by:
ηt=φt+ξt Cov (φt, φt−m) =ρmσφ2
(3.17)
where ρ is the AR parameter that controls the dependence of the noise, φ and ξ are independent with zero means and third moments, ξt is i.i.d., E (ξ2) =σξ2, E (φ2) =σφ2,
E (φ4) = κ
φσ4φ, E (ξ4) =κξσξ4. Observed returns are then defined as:
rt,h =p∗t −p∗t−h+ηt−ηt−h =r∗t,h+et,h
et,h =φt−φt−h+ξt−ξt−h ≡∆mφt+ ∆mξt
(3.18)
First of all note that since we construct all our realized volatility estimators using intradaily prices, i.e. we do not take into account overnight returns, only daily variances will be changed because of the non-i.i.d.noise. In contrast, the covariance structure of the realized volatility estimators should not change. Consequently, given the dependent microstructure noise process, we have that:
Theorem 3.3.3 Let Assumption 3.3.2 hold, then the variance for the “plain vanilla” realized volatility estimators is:
Var(RVjm) = 2 p X i=1 a2 i λ2 i e−λiM h−1 +λ iM h + 2Ma20h2 + 4M p X i=1 a2 i λ2 i e−λih−1 +λ ih + 8a0hM((1−ρm)σφ2+σξ2) +(3.19) where Var M X k=1 e(j/Mm)2+kh,m =MVar e(m)2+ 2(M −1)Var ξt2−m+ 4(1−ρm)2σ2φσξ2 + 2Cov {φt−φt−m}2,{φt−m−φt−2m}2 MX−1 i=1 (M −i)ρ2m(i−1) (3.19)
which also appears as equation (A.43) in Appendix Appendix A.2, where further details
are provided. Moreover, an approximate optimal sampling is expressed as: MRV 'arg min 2Q M −8a0ρ mσ2 φ+(3.19)
where m=M/M and equation (3.19) evaluates the variance of the market microstruc- ture noise part.
Proof: see Appendix A.2.
To understand the impact of dependent noise on the optimal sampling frequency consider expression (3.19) in the above Theorem. Since one can control for many parameter variations, we will focus on only three, namely the number of observations in the subsampling gridM, the dependence levelρand the relative variance magnitude between σ2
ξ and σ2φ. In fact, we can divide the whole range of M into two regions: low-frequency region, for which we can assume that the market microstructure noise is i.i.d. and high-frequency region where the dependence appears. Thus, we observe differences from the i.i.d. case if:
• ρ6= 0
• We are in the reasonably high-frequency region (ρm 6= 0)
• The variance of the φt is larger than or comparable with the variance of ξt
The last requirement is due to the well-known fact that the sum of AR(1) and white noise processes is ARMA(1,1) and as σ2
ξ/σ2φ→ ∞, MA(1) coefficient converges to −ρ, which makes the resulting noise process i.i.d.
If any of these conditions does not hold, we can effectively assume that ρ = 0. We obtain then the i.i.d. case with σ2
η = σ2ξ + σφ2 and κ = (Var (ξ2) + Var (φ2) + 4σ2
ξσφ2)/(σξ2+σφ2)2. In this case the variance increase associated with the microstructure
noise term (3.17) is linear with respect toM, and the variance decrease associated with the discretization noise is proportional to 1/M.
When ρ >0, Var (et,h) = 2σξ2+ 2σφ2(1−ρm)<2(σξ2+σφ2), and∂Var (et,h)/∂M <0. At the extreme, when ρ = 1, Var (et,h) = 2σξ2. As a result, for positive ρ, the optimal frequency should be higher compared to thei.i.d. case.
The situation is more complicated whenρ <0.Depending on whether the powerm
is even or odd (which changes the sign of ρm), the Var (et,h) can be greater or smaller than the variance of the noise under the i.i.d. assumption. As a result, the “plain vanilla”RV variance is no longer smooth and the optimal sampling frequency is harder to compute.
To illustrate our findings, consider Figure 3.3. In this figure we plot the variance of the market microstructure noise (equation 3.19) and the variance of the “plain vanilla” realized volatility (A.47) on the Y-axis againstM for three different parameter settings of ρ = 0., −0.9 and .9 and two ratios σ2
φ/σξ2 = 1,10. We note from Figure 3.3 that as ρ increases, the variance increases slower as function of M compared to the ρ = 0 case. In fact, the variance of the noise for the case ofρ = .9 and σ2
φ/σξ2 = 10 decreases with the increase in M. Therefore, since the discretization noise decreases as sampling frequency increases, there is no internal minimum and the “plain vanilla”RV variance achieves its minimum at the highest possible frequency.