option pricing - Consensus on Cauchy distribution for stock prices

What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.

My motivation is to find a distribution for the stochastic process governing infinitesimally small stock price movements $\Delta W_t$. The standard process used is the Wiener process depending on a normal random variable $\epsilon$ i.e. $\Delta W_t = \epsilon \sqrt{t}$. This results in the problem that resulting prices are normally distributed, but it is well known that stock prices have heavier tails than that.

In fact it seems that if $\epsilon$ follows any finite variance distribution, it will result in normally distributed prices by the CLT.

I am therefore looking for a stable distribution to model stock prices and the Cauchy immediately came to mind.