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The impact of asynchronous trading on Epps effect on Warsaw Stock Exchange

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Abstract

The main goal of the analysis is the verification of whether asynchrony in transaction times is a considerable cause of the Epps effect on the Warsaw Stock Exchange among the most liquid assets. A method for compensating for the impact of asynchrony in trading on the Epps effect is presented. The method is easily applicable. Calculations are made using the exact time of transactions and prices of the assets. The estimation is not biased by intervals during which no transactions have taken place. Among all the analyzed stock pairs, asynchrony turns out to be the main cause of the Epps effect. However, the corrected correlation estimator seems to be more volatile than the regular estimator of the correlation. The presented analysis can be reproduced for the same data or replicated for another dataset; all R codes used in the process of writing this article are available upon request. The main novelty/value added of this paper is the application to an emerging market of a new method for compensating for asynchrony in trading.

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Acknowledgments

Financial support for this paper from the National Science Centre of Poland (Research Grant DEC-2012/05/B/HS4/00810) is gratefully by Henryk Gurgul acknowledged. Financial support for this paper from the Dean of Faculty of Management, AGH University (Statutory Activity No. 15/11.200.296) is gratefully acknowledged by Artur Machno. We would like to thank the two anonymous referees for their valuable comments on an earlier version of the paper.

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Authors and Affiliations

  1. Department of Applications of Mathematics in Economics, Faculty of Management, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Kraków, Poland

    Henryk Gurgul & Artur Machno

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  1. Henryk Gurgul
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  2. Artur Machno
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Correspondence to Henryk Gurgul.

Appendix

Appendix

Proof of Theorem 4.1

Let us first calculate the observed normalized process \(g_{\Delta t}^i (t)\) in terms of the underlying process \(\tilde{g} ^{i}(t)\), where \(i=1,2\). The operator \(\langle \cdots \rangle \) means the average over the analyzed period \(\left[ {0,\hbox {T}}\right] \).

$$\begin{aligned} g_{\Delta t}^i (t)= & {} \frac{{r}_{\Delta t}^i (t)-\langle {r}_{\Delta t}^i\rangle }{\sigma _{\Delta t}^i }=\frac{\mathop \sum \nolimits _{j=1}^{N_{\Delta t}^i (t)}\tilde{r} ^{i}\left( {\gamma ^{i}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) -\langle {r}_{\Delta t}^i \rangle }{\sigma _{\Delta t}^i }\\= & {} \frac{\mathop \sum \nolimits _{j=1}^{N_{\Delta t}^i (t)} \left( {\tilde{\sigma }^{i}\tilde{g} ^{i}\left( {\gamma ^{i}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) +\langle \tilde{r}^{i}\rangle } \right) -\langle {r}_{\Delta t}^i\rangle }{\sigma _{\Delta t}^i}\\= & {} \frac{\tilde{\sigma }^{i}\mathop \sum \nolimits _{j=1}^{N_{\Delta t}^i (t)} \tilde{g}^{i}\left( {\gamma ^{i}\left( \hbox {t} \right) +j\Delta \tilde{t}}\right) +N_{\Delta t}^i (t)\langle \tilde{r}^{i}\rangle -\langle {r}_{\Delta t}^i \rangle }{\sigma _{\Delta t}^i} \end{aligned}$$

Under Assumption A1:

$$\begin{aligned} \langle {r}_{\Delta t}^i (t)\rangle =\langle {N}_{\Delta t}^i\rangle \langle \tilde{r}^{i}\rangle ;\left( {\sigma _{\Delta t}^i } \right) ^{2}=\langle {N}_{\Delta t}^i \rangle \left( {\tilde{\sigma }^{i}} \right) ^{2} \end{aligned}$$

Thus,

$$\begin{aligned} g_{\Delta t}^i (t)=\frac{1}{\sqrt{\langle {N}_{\Delta t}^i \rangle }}\mathop \sum \limits _{j=1}^{N_{\Delta t}^i (t)} \tilde{g}^{i}\left( {\gamma ^{i}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) +\frac{\langle \tilde{r}^{i}\rangle \left( {N_{\Delta t}^i (t)-\langle {N}_{\Delta t}^i \rangle } \right) }{\sigma _{\Delta t}^i } \end{aligned}$$

Let us now calculate the observed correlation \(\hbox {corr}\left( {r_{\Delta t}^1 ,r_{\Delta t}^2 } \right) \) in terms of the correlation of underlying processes \(\hbox {corr}_{t_j } \left( {\tilde{g}^{1},\tilde{g}^{2}} \right) \).

$$\begin{aligned}&\hbox {corr}\left( {r_{\Delta t}^1 ,r_{\Delta t}^2 } \right) =\frac{1}{T}\mathop \sum \limits _{j=1}^T g_{\Delta t}^1 (t)g_{\Delta t}^2 (t)\\&\quad =\frac{1}{T}\mathop \sum \limits _{j=1}^T \left( \left( {\frac{1}{\sqrt{\langle {N}_{\Delta t}^1\rangle }}\mathop \sum \limits _{j=1}^{N_{\Delta t}^1 (t)} \tilde{g}^{1}\left( {\gamma ^{1}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) +\frac{\langle \tilde{r}^{1}\rangle \left( {N_{\Delta t}^1 (t)-\langle {N}_{\Delta t}^1 \rangle } \right) }{\sigma _{\Delta t}^1}} \right) \right. \\&\qquad \left. \times \left( {\frac{1}{\sqrt{\langle {N}_{\Delta t}^2\rangle }}\mathop \sum \limits _{j=1}^{N_{\Delta t}^2 (t)} \tilde{g}^{2}\left( {\gamma ^{2}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) +\frac{\langle \tilde{r}^{2}\rangle \left( {N_{\Delta t}^2 (t)-\langle {N}_{\Delta t}^2\rangle } \right) }{\sigma _{\Delta t}^2 }} \right) \right) \end{aligned}$$

The average of the second component in each factor equals zero, thus under Assumption A1 we have.

$$\begin{aligned} \hbox {corr}\left( {r_{\Delta t}^1 ,r_{\Delta t}^2 } \right)= & {} \frac{1}{T}\mathop \sum \limits _{j=1}^T \left( \frac{1}{\sqrt{\langle {N}_{\Delta t}^1 \rangle }}\mathop \sum \limits _{j=1}^{N_{\Delta t}^1 (t)} \tilde{g} ^{1}\left( {\gamma ^{1}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) \right. \\&\left. \times \frac{1}{\sqrt{\langle {N}_{\Delta t}^2 \rangle }}\mathop \sum \limits _{j=1}^{N_{\Delta t}^2 (t)} \tilde{g} ^{2}\left( {\gamma ^{2}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) \right) \end{aligned}$$

The means of the process \(\tilde{g}^{1}\) and \(\tilde{g}^{2}\) equal zero, thus under Assumption A2 we have.

$$\begin{aligned}&\hbox {corr}\left( {r_{\Delta t}^1 ,r_{\Delta t}^2 } \right) \\&\quad =\frac{1}{T}\mathop \sum \limits _{j=1}^T \left( {\frac{1}{\sqrt{\langle N_{\Delta t}^1\rangle \langle N_{\Delta t}^2\rangle }}\mathop \sum \limits _{j=1}^{\overline{N}\left( {t_j } \right) } \tilde{g}^{1}\left( {\gamma ^{1}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) \tilde{g}^{2}\left( {\gamma ^{2}\left( \hbox {t} \right) +j\Delta \tilde{t}} \right) } \right) \\&\quad =\frac{1}{T}\mathop \sum \limits _{j=1}^T \left( \frac{ \overline{N} \left( {t_j } \right) }{\sqrt{\langle N_{\Delta t}^1\rangle \langle N_{\Delta t}^2 \rangle }}\hbox {corr}_{t_j } \left( {\tilde{g}^{1},\tilde{g}^{2}} \right) \right) \end{aligned}$$

Thus,

$$\begin{aligned} \hbox {corr}_c \left( {r_{\Delta t}^1 ,r_{\Delta t}^2 } \right) =\frac{1}{T}\mathop \sum \limits _{j=1}^T g_{\Delta t}^1 (t)g_{\Delta t}^2 (t)\frac{\sqrt{\langle N_{\Delta t}^1\rangle \langle N_{\Delta t}^2\rangle }}{\overline{N}\left( {t_j } \right) }; \end{aligned}$$

which had to be proven. \(\square \)

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Gurgul, H., Machno, A. The impact of asynchronous trading on Epps effect on Warsaw Stock Exchange. Cent Eur J Oper Res 25, 287–301 (2017). https://doi.org/10.1007/s10100-016-0442-y

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