  In the past two decades, the field of applied finance has tremendously
benefited from graph theory. As a result, novel methods ranging from asset
network estimation to hierarchical asset selection and portfolio allocation are
now part of practitioners' toolboxes. In this paper, we investigate the
fundamental problem of learning undirected graphical models under Laplacian
structural constraints from the point of view of financial market times series
data. In particular, we present natural justifications, supported by empirical
evidence, for the usage of the Laplacian matrix as a model for the precision
matrix of financial assets, while also establishing a direct link that reveals
how Laplacian constraints are coupled to meaningful physical interpretations
related to the market index factor and to conditional correlations between
stocks. Those interpretations lead to a set of guidelines that practitioners
should be aware of when estimating graphs in financial markets. In addition, we
design numerical algorithms based on the alternating direction method of
multipliers to learn undirected, weighted graphs that take into account
stylized facts that are intrinsic to financial data such as heavy tails and
modularity. We illustrate how to leverage the learned graphs into practical
scenarios such as stock time series clustering and foreign exchange network
estimation. The proposed graph learning algorithms outperform the
state-of-the-art methods in an extensive set of practical experiments.
Furthermore, we obtain theoretical and empirical convergence results for the
proposed algorithms. Along with the developed methodologies for graph learning
in financial markets, we release an R package, called fingraph, accommodating
the code and data to obtain all the experimental results.
