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We propose novel randomized geometric tools to detect low-volatilityanomalies in stock markets; a principal problem in financial economics. Ourmodeling of the (detection) problem results in sampling and estimating the(relative) volume of geodesically non-convex and non-connected sphericalpatches that arise by intersecting a non-standard simplex with a sphere. Tosample, we introduce two novel Markov Chain Monte Carlo (MCMC) algorithms thatexploit the geometry of the problem and employ state-of-the-art continuousgeometric random walks (such as Billiard walk and Hit-and-Run) adapted onspherical patches. To our knowledge, this is the first geometric formulationand MCMC-based analysis of the volatility puzzle in stock markets. We haveimplemented our algorithms in C++ (along with an R interface) and we illustratethe power of our approach by performing extensive experiments on real data. Ouranalyses provide accurate detection and new insights into the distribution ofportfolios' performance characteristics. Moreover, we use our tools to showthat classical methods for low-volatility anomaly detection in finance form badproxies that could lead to misleading or inaccurate results.

Randomized geometric tools for anomaly detection in stock markets