A possible interpretation of financial markets affected by dark volatility

The aim of this paper is to use a special type of Einstein warped product manifolds recently introduced, the so-called PNDP-manifolds, for the differential geometric study, by focusing on some aspects related to dark field in financial market such as the concept of dark volatility. This volatility is not fixed in any relevant economic parameter, a sort of negative dimension, a ghost field, that greatly influences the behavior of real market. Since the PNDP-manifold has a "virtual" dimension, we want to use it in order to show how the Global Market is influenced by dark volatility, and in this regard we also provide an example, by considering the classical exponential models as possible solutions to our approach. We show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.