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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 the approach investigated in \cite{Romero}, 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.

Wild Randomness and the Application of Hyperbolic Diffusion in Financial Modelling