  We propose novel randomized geometric tools to detect low-volatility
anomalies in stock markets; a principal problem in financial economics. Our
modeling of the (detection) problem results in sampling and estimating the
(relative) volume of geodesically non-convex and non-connected spherical
patches that arise by intersecting a non-standard simplex with a sphere. To
sample, we introduce two novel Markov Chain Monte Carlo (MCMC) algorithms that
exploit the geometry of the problem and employ state-of-the-art continuous
geometric random walks (such as Billiard walk and Hit-and-Run) adapted on
spherical patches. To our knowledge, this is the first geometric formulation
and MCMC-based analysis of the volatility puzzle in stock markets. We have
implemented our algorithms in C++ (along with an R interface) and we illustrate
the power of our approach by performing extensive experiments on real data. Our
analyses provide accurate detection and new insights into the distribution of
portfolios' performance characteristics. Moreover, we use our tools to show
that classical methods for low-volatility anomaly detection in finance form bad
proxies that could lead to misleading or inaccurate results.
