When I did econometric studies on Options, I did a test on the distribution of price changes in many various stocks, and the results did not approximate the Black Scholes Model exactly. In fact, the distribution of sigma changes in stock changes is not lognormal at all -- there are many high 3.5 - 4.0 sigma plus moves in stocks.
Actually, at the end of my study, I found that, if one were to precisely use the Black Scholes model to find out if writing an option would be profitable, many times the option writer would get screwed.
Also, there is an equation that is used to determine if a stock price can surpass a certain minimum or maximum, but that standard equation, which is used in the Black Scholes model, is flawed. It basically just checks if the stock price can get to a certain point *AT* expiration. There is a much more refined equation that actually shows that a stock can get to price X *anytime* before expiration.
I also did some work with delta-normal position strategies, and using options for VEGA (tau) strategies by normalizing the delta and playing options just off volatility changes.
There are old-timers that go so far as finding the gamma of gamma, but my own research has shown that this is just a waste of time.
However, if someone has a winning strategy, all the math in the world is not going to prove them wrong.