Why so negative on negative volatilities

Abstract

Most financial models are mathematically rigorously formulated using continuous time. However, asset prices in reality appear in discrete time intervals. Hence, there is the need to discretize financial models. During the process of discretization, stochastic volatilities can get negative. The typical way of dealing with this problem is setting these negative volatilities to zero. This is arbitrary and conceptually inconsistent. We argue that it is a better solution to accept negative volatilities and integrate them into the model. In this paper, we define negative volatilities, give examples how they occur in financial modeling and derive a mathematical theory of negative volatilities.