Wild Randomness, and the application of Hyperbolic Diffusion in Financial Modelling

TL;DR

This paper explores how hyperbolic diffusion and quantum probability can be used to construct non-Gaussian Martingales with Cauchy distribution marginals, offering new insights into modeling extreme financial events.

Contribution

It introduces a novel method for constructing Martingale processes with Cauchy distribution marginals using hyperbolic diffusion and quantum probability techniques.

Findings

Martingale processes with Cauchy marginals are constructed.

Links between hyperbolic diffusion and financial modeling are established.

Quantum probability provides a framework for non-Gaussian Martingales.

Abstract

The application of the Cauchy distribution has often been discussed as a potential model of the financial markets. In particular the way in which single extreme, or "Black Swan", events can impact long term historical moments, is often cited. In this article we show how one can construct Martingale processes, which have marginal distributions that tend to the Cauchy distribution in the large volatility limit. This provides financial justification to approaches discussed by other authors, and highlights an example of how quantum probability can be used to construct non-Gaussian Martingales. We go on to illustrate links with hyperbolic diffusion, and discuss the insight this provides.