  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.
