Disentangling and quantifying market participant volatility contributions

Abstract

Thanks to the access to labeled orders on the CAC 40 index future provided by Euronext, we are able to quantify market participants contributions to the volatility in the diffusive limit. To achieve this result, we leverage the branching properties of Hawkes point processes. We find that fast intermediaries (e.g. market maker type agents) have a smaller footprint on the volatility than slower, directional agents. The branching structure of Hawkes processes allows us to examine also the degree of endogeneity of each agent behavior, and we find that high-frequency traders are more endogenously driven than other types of agents.

Keywords:

• Volatility
• Hawkes process
• Agent-based model
• High-frequency data
• Agent behavior

Acknowledgments

We thank Euronext for making their data available to us. We also thank Laurent Fournier, Angelique Begrand and Luxi Chen for useful discussions. This research benefited from the support of the chair of the Risk Foundation: Quantitative Management Initiative.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† Despite some non-zero mean price return can be observed at long term, notably on stock markets, this trend is negligible at the microstructure scale, so one can safely assume it is zero.

‡ In fact we must assume that $C^{ij}\left(z\right)$ is integrable, i.e. that $C^{ij}\left(z\right)$ decreases faster than $z^{-\gamma }$ with $\gamma >1$. In that case, ${\int }_0^{\tau }y\left[C\right(y)+C(-y\left)\right]\mathrm{d}y=O\left({\tau }^{2-\gamma }\right)=o\left(\tau \right)$ and we have the expected result. Empirically, despite long-range dependence has been observed in the dynamics of some book events, we consider the time scale τ large enough (typically few minutes are sufficient for our purpose) such that this regime no longer holds (see e.g. Bacry et al. 2016).

† The choice of the functions is typically dictated by the type of application, in financial applications the usual choice is given by exponential functions.

‡ Indicating with λ the ground truth intensity and with ${\lambda }_{\theta }$ the parametric version, minimizing the contrast function (17(17) ${\mathcal{C}}^{i,\alpha }=T^{-1}{\int }_0^T{\left({\lambda }_s^{i,\alpha }\right)}^2\mathrm{d}s-\frac{2}{T}{\int }_0^T{\lambda }_s^{i,\alpha }\mathrm{d}{N}_s^{i,\alpha }.$(17) ) is equivalent to minimizing $\mathbb{E}[\parallel {\lambda }_{\theta }-\lambda {\parallel }^2]$ where $\parallel f{\parallel }^2=\int \mathrm{\infty \infty }{f}^2\left(t\right)\mathrm{d}t$ (see e.g. Hansen et al. 2015, Bacry and Muzy 2016)

§ With the exception of orders refused by the market upon submission, e.g. Fill Or Kill orders that are not filled.

¶ For the CAC 40 future there are 3 monthly, 3 quarterly (from March, June, September, December), and 8 half-yearly maturities from the June/December cycle.

† Trading ceases at 16:00 CET on the third Friday of the delivery month. In the event that the third Friday is not a business day, the Last Trading Day shall normally be the last business day preceding the third Friday.

† The choice of the average is natural as $\mathrm{\Delta }P=\sum _{k=1,2,\dots ,}{\delta }_k\mathrm{\Delta }{N}^k=\mathrm{\Delta }N\sum _{k=1,2,\dots ,}\frac{{\delta }_k\mathrm{\Delta }{N}^k}{\mathrm{\Delta }N}=\mathrm{\Delta }N\overline{\delta }$.

‡ We tried several combinations of the hyperparameters and we found this one to represent the best compromise in terms of number of parameters and good description of the data. The results presented in the following sections are however quite stable over the set of hyper parameters we tried.

§ To convert the Hawkes squared volatility estimation ${\sigma }_H^2$ calculated as in (12(12) $\begin{array}{rl}\frac{{\sigma }_{\tau }^2}{\tau }\to \sum _{i,j\in \mathcal{M}}{\delta }_i{\delta }_j{C}^{ij}& =\sum _{m=1}^n{\mathrm{\Lambda }}^m\sum _{i,j\in \mathcal{M}}{\delta }_i{\delta }_j{R}^{im}{R}^{jm}\\ & =\sum _{m=1}^n{\mathrm{\Lambda }}^m{\left(\sum _{i\in \mathcal{M}}{\delta }_i{R}^{im}\right)}^2.\end{array}$(12) ) to the annualized value we use the formula $\sigma =\sqrt{{\sigma }_H^2\ast T}\cdot \frac{S}{P_0},$ where S=0.25 is the half-tick size, $P_0$ is the open price of the day, and $T=8.5\cdot 3600\cdot 252$ is the factor to annualize.

† Notice that, in strictly speaking, the matrix $R^{i,\alpha ;j,\beta }$ for $i,j\ne m$ depends on agent m features and one should recompute a new $\mathbf{R}$ matrix without this agent contribution. However it is easy to show that generically, the influence of m on the matrix $\mathbf{R}$ is a second order perturbation, i.e. of relative magnitude $\sim M^{-2}$. Given that M=17, we can safely consider that this has a negligible impact (of the order of 0.3%) as compared to all sources of errors.

† All the comments we do here rely on a direct interpretation of the empirical plots. Some degree of caution is thus needed since a statistical analysis would be necessary in order to rigorously differentiate noise from outliers.