\( \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \newcommand{\textnormal}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}} \)

5.6 Summary

While the topic of estimation of graphical models goes back at least to the 1970s, the first application in financial markets can only be found in the seminal paper of Mantegna (1999), where a simple correlation graph was employed. Since then, a myriad methods have been proposed as surveyed in Marti et al. (2021).

Among the many methods described in this chapter, only a few seem to be appropriate for financial time series and able to produce desirable graphs:

• Sparse GMRF graphs (Section 5.2.3) as formulated in (5.7) are a good start.

• Asset clustering is a data-driven alternative to handcrafted sectors or industries. For this purpose, \(k\)-component graphs (Section 5.3.1) can be readily obtained by imposing low-rank structure with degree control as in formulation (5.9).

• Lastly, owing to the inherent heavy-tailed nature of financial data, heavy-tailed graph models (Section 5.4) are undoubtedly more suitable than Gaussian ones. These models are formulated in (5.13) and can be practically solved iteratively using the simpler problems outlined in (5.14).

References

Mantegna, R. N. (1999). Hierarchical structure in financial markets. The European Physical Journal B – Condensed Matter and Complex Systems, 11, 193–197.

Marti, G., Nielsen, F., Binkowski, M., and Donnat, P. (2021). A review of two decades of correlations, hierarchies, networks and clustering in financial markets. In F. Nielsen, editor, Progress in information geometry, pages 245–274. Springer.