Financial and risk modelling with semicontinuous covariances

Abstract

We are encountering deep financial crisis. One of the basic problems we face is how to predict financial developments. Current modelling is clearly insufficient since continuity of autocorrelation oversmoothes the empirical financial data. We prove that geometric probability to chose positive definite continuous covariance by financial data is zero. We also have the empirical evidence for this fact from real stock data. In order to overcome this gap we introduce the novel abc class of semicontinuous covariance functions, which integrates both continuous and discontinuous covariances. The abc class of covariances is much more flexible, powerful, and realistic than its continuous counterpart. It gives answer to the question why financial markets face negative interest rates, cyclic downbreaks, and other difficulties nowadays. As it is clearly illustrated in this paper, the experimenter (e.g. the financial institution) can choose only up to some extent whether abc class member will be positive definite everywhere. To the best knowledge of the authors this fits well to the own Kolmogorov’s opinions on axiomatic theory of probability. The Kolmogorov’s axiomatic theory of probability was developed as one possible alternative in order to utilize the flexibility of Lebesgue integral in probability theory, but in many experiments e.g. in physics and finance, one should go beyond this axiomatics. That means one should accept negative probabilities (and consequently negative definite covariance functions). In this paper we clarify that the modeler’s choice of positive/negative definite functions should be considered within broader Information Theory setup. The introduced abc class is purely topologically defined, with soft regularity conditions on covariances, which are still applicable for increasing and infill domain asymptotics for regression problems, kriging, and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g. by earthquakes). In particular, we construct a random walk model with semicontinuous covariance. The novel abc class provides a proper environment for unifying several contradictions on IS/ML model in the economic literature. All mentioned novel economical applications are hardly visible from currently used “continuous covariance” framework.