Spot Volatility Estimation for High-Frequency Data: Adaptive Estimation in Practice

Till Sabel; Johannes Schmidt-Hieber; Axel Munk

doi:10.1007/978-3-319-18732-7_12

Modeling and Stochastic Learning for Forecasting in High Dimensions pp 213-241 | Cite as

Spot Volatility Estimation for High-Frequency Data: Adaptive Estimation in Practice

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Till Sabel
Johannes Schmidt-Hieber
Axel Munk

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Part of the Lecture Notes in Statistics book series (LNS, volume 217)

Abstract

We develop further the spot volatility estimator introduced in Hoffmann et al. (Ann Inst H Poincaré (B) Probab Stat 48(4):1186–1216, 2012) from a practical point of view and make it applicable to the analysis of high-frequency financial data. In a first part, we adjust the estimator substantially in order to achieve good finite sample performance and to overcome difficulties arising from violations of the additive microstructure noise model (e.g. jumps, rounding errors). These modifications are justified by simulations. The second part is devoted to investigate the behavior of volatility in response to macroeconomic events. We give evidence that the spot volatility of Euro-BUND futures is considerably higher during press conferences of the European Central Bank. As an outlook, we present an estimator for the spot covolatility of two different prices.

Keywords

Wavelet Coefficient European Central Bank Microstructure Effect Mean Integrate Square Error Heston Model

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Notes

Acknowledgements

Support of DFG/SNF-Grant FOR 916, DFG postdoctoral fellowship SCHM 2807/1-1, and Volkswagen Foundation is gratefully acknowledged. We appreciate the help of CRC 649 “Economic Risk” for providing us with access to Eurex database. Parts of this work are taken from the PhD thesis Schmidt-Hieber [48]. We thank Marc Hoffmann, Markus Reiß, Markus Bibinger, and an anonymous referee for many helpful remarks and suggestions.

Appendix A: Proof of Lemma 1

The proof is the same as in the PhD thesis Schmidt-Hieber [48]. We include it for sake of completeness.To keep notation simple, we use the following quantitiesin the spirit of the definitions of Sect. 2.3: For any process(A _i, n) ∈ { (Y _i, n), (ε _i, n), (X)_i, n}, define

$$\displaystyle{ \overline{A}_{i,m} = \overline{A}_{i,m}(\lambda ):= \frac{m} {n} \sum _{\frac{j} {n}\in [\frac{i-2} {m}, \frac{i} {m}]}\lambda \big(m\tfrac{j} {n} - (i - 2)\big)A_{j,n}. }$$

$$\displaystyle{ \mathfrak{b}(A)_{i,m} = \mathfrak{b}(\lambda,A_{\cdot })_{i,m}:= \frac{m^{2}} {2n^{2}}\sum _{ \frac{j} {n}\in [\frac{i-2} {m}, \frac{i} {m}]}\lambda ^{2}\big(m\tfrac{j} {n} - (i - 2)\big)\big(A_{j,n} - A_{j-1,n}\big)^{2}. }$$

Further, recall that our estimator for the integrated volatility is given by $\widehat{\langle 1,\sigma ^{2}\rangle } =\sum _{ i=2}^{m}\overline{Y }_{i,m}^{2} - \mathfrak{b}(Y )_{i,m}$.

To prove the lemma, let us first show that the bias is of smaller order than n ^−1∕4. In fact, note that $\mathbb{E}\big[\ \overline{Y }_{i,m}^{2}\big] = \mathbb{E}\big[\ \overline{X}_{i,m}^{2}\big] + \mathbb{E}\big[\overline{\epsilon }_{i,m}^{2}\big].$ Clearly, one can bound

$$\displaystyle{ \Big\vert \mathbb{E}\big[\overline{\epsilon }_{i,m}^{2}\big] - \mathbb{E}\big[\mathfrak{b}(\lambda,Y )_{ i,m}\big]\Big\vert = O( \tfrac{1} {n}). }$$

Further, Lipschitz continuity of λ together with a Riemann approximation argument gives us

$$\displaystyle{ \big\vert \mathbb{E}\big[\ \overline{X}_{i,m}^{2}\big] - \tfrac{\sigma ^{2}} {m}\big\vert =\Big \vert \tfrac{\sigma ^{2}} {m}\int _{0}^{2}\int _{ 0}^{2}\lambda (s)\lambda (t)(s \wedge t)dtds - \tfrac{\sigma ^{2}} {m}\Big\vert + O( \tfrac{1} {n}) = O( \tfrac{1} {n}). }$$

Here, the last equation is due to partial integration and the definition of a pre-average function (cf. Definition 2.1). Since both approximations are uniformly in i, this shows that the bias is of order O(n ^−1∕2).

For the asymptotic variance, first observe that $V ar(\sum _{i=2}^{m}\mathfrak{b}(\lambda,Y )_{i,m}) = o(n^{-1/2}).$ Hence,

$$\displaystyle{V ar(\widehat{\langle 1,\sigma ^{2}\rangle }) = V ar(\sum _{ i=2}^{m}\overline{Y }_{ i,m}^{2}) + o\Big(n^{-1/4}\big(V ar(\sum _{ i=2}^{m}\overline{Y }_{ i,m}^{2})\big)^{1/2} + n^{-1/2}\Big),}$$

by Cauchy-Schwarz inequality. Recall that for centered Gaussian random variables U and V, Cov(U ², V ²) = 2(Cov(U, V ))². Therefore, it suffices to compute $Cov(\overline{Y }_{i,m},\overline{Y }_{k,m}) = \mathbb{E}[\overline{Y }_{i,m}\overline{Y }_{k,m}]$.

By the same arguments as above, that is Riemann summation and partial integration, we find

$$\displaystyle{ \mathbb{E}\Big[\Big\vert \overline{X}_{i,m}\overline{X}_{k,m} -\int _{0}^{1}\varLambda (ms - (i - 2))dX_{ s}\int _{0}^{1}\varLambda (ms - (k - 2))dX_{ s}\Big\vert \Big] \lesssim n^{-1}. }$$

Therefore,

$$\displaystyle{ \mathbb{E}\big[\overline{X}_{i,m}\overline{X}_{k,m}\big] =\sigma ^{2}\int _{ 0}^{1}\varLambda (ms - (i - 2))\varLambda (ms - (k - 2))ds + O(n^{-1}), }$$

where the last two arguments hold uniformly in i, k.

In order to calculate $\mathbb{E}[\overline{Y }_{i,m}\overline{Y }_{k,m}],$ we must treat three different cases, | i − k | ≥ 2, | i − k | = 1 and i = k, denoted by I, II and III.

ad I. In this case $(\tfrac{i-2} {m}, \tfrac{i} {m}]$ and $(\tfrac{k-2} {m}, \tfrac{k} {m}]$ do not overlap. By the equalities above, it follows $Cov(\overline{Y }_{i,m},\overline{Y }_{k,m}) = O(n^{-1}).$

ad II. Without loss of generality, we set k = i + 1. Then, we obtain

$$\displaystyle\begin{array}{rcl} & & Cov(\overline{Y }_{i,m},\overline{Y }_{i+1,m}) = \mathbb{E}\big[\overline{X}_{i,m}\overline{X}_{i+1,m}\big] + \mathbb{E}\big[\overline{\epsilon }_{i,m}\overline{\epsilon }_{i+1,m}\big] {}\\ & =& \sigma ^{2}\int _{ 0}^{1}\varLambda (ms - (i - 2))\varLambda (ms - (i - 1))ds + O(n^{-1}) {}\\ & & +\tau ^{2}\frac{m^{2}} {n^{2}} \sum _{ \tfrac{j} {n} \in \big(\tfrac{i-2} {m}, \tfrac{i} {m}\big]}\lambda (m\tfrac{j} {n} - (i - 2))\lambda (m\tfrac{j} {n} - (i - 1)) {}\\ & =& \frac{\sigma ^{2}} {m}\int _{0}^{1}\varLambda (u)\varLambda (1 + u)du +\tau ^{2}\frac{m} {n} \int _{0}^{1}\lambda (u)\lambda (1 + u)du + O(n^{-1}), {}\\ \end{array}$$

where the last inequality can be verified by Riemann summation. Noting that λ is a pre-average function, we obtain λ(1 + u) = −λ(1 − u) and

$$\displaystyle\begin{array}{rcl} & Cov(\overline{Y }_{i,m},\overline{Y }_{i+1,m}) = \frac{\sigma ^{2}} {m}\int _{0}^{1}\varLambda (u)\varLambda (1 - u)du -\frac{\tau ^{2}m} {n} \int _{0}^{1}\lambda (u)\lambda (1 - u)du + O(n^{-1}).& {}\\ \end{array}$$

ad III. It can be shown by redoing the arguments in II that

$$\displaystyle\begin{array}{rcl} & V ar(\overline{Y }_{i,m}) = V ar(\overline{X}_{i,m}) + V ar(\overline{\epsilon }_{i,m}) = \frac{\sigma ^{2}} {m}\int _{0}^{2}\varLambda ^{2}(u)du +\tau ^{2}\frac{m} {n} \int _{0}^{2}\lambda ^{2}(u)du + O(n^{-1}).& {}\\ \end{array}$$

Note that $\|\varLambda \|_{L^{2}[0,2]} = 1.$ Since the above results hold uniformly in i, k, it follows directly that

$$\displaystyle\begin{array}{rcl} & & V ar(\sum _{i=2}^{m}\overline{Y }_{ i,m}^{2}) {}\\ & =& \sum _{i,k=2,\ \vert i-k\vert \geq 2}^{m}2\big(Cov(\overline{Y }_{ i,m},\overline{Y }_{k,m})\big)^{2} {}\\ & & +2\sum _{i=2}^{m-1}2\big(Cov(\overline{Y }_{ i,m},\overline{Y }_{i+1,m})\big)^{2} +\sum _{ i=2}^{m}2\big(V ar(\overline{Y }_{ i,m})\big)^{2} {}\\ & =& O(n^{-1}) + 4\Big( \frac{\sigma ^{2}} {\sqrt{c}}\int _{0}^{1}\varLambda (u)\varLambda (1 - u)du -\tau ^{2}c^{3/2}\int _{ 0}^{1}\lambda (u)\lambda (1 - u)du\Big)^{2}n^{-1/2} {}\\ & & +2\Big( \frac{\sigma ^{2}} {\sqrt{c}} + 2\tau ^{2}c^{3/2}\|\lambda \|_{ L^{2}[0,1]}^{2}\Big)^{2}n^{-1/2}. {}\\ \end{array}$$

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Till Sabel
- 1
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Johannes Schmidt-Hieber
- 2
Axel Munk
- 1

1.Department of Mathematics and Computer Science, Institute for Mathematical StochasticsGeorg-August-University GöttingenGöttingenGermany
2.University of LeidenLeidenNetherlands

Spot Volatility Estimation for High-Frequency Data: Adaptive Estimation in Practice

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