What do you mean what should we do? Why do you have to do anything.... not sure what kind of question you are asking.
Not sure the purpose of using BSM, the option price is already observed in the market. So what would you be using it for? IV is also shown in the market. If you want to put your own volatility input that there are hundreds of ways to come up with a flawed volatility estimate so will serve no purpose. Not sure why you need to even use BSM at all unless you are trying to price a privately negotiated OTC option.
In Sept 2015, GOOGL was trading in a range around $630 the IV at that time was about 10 points higher than historical (~40%) one could short a 90 days 1-sigma OTM call and collected a nice premium with only a 15% probability of being called according to BSM and IV. GOOGL went up by over 100 points by mid Oct and stayed there. So that trade would be a loser. This is an example of using BSM to get the 1-sigma OTM strike price and calculated the expected return for this trade. If one uses a fat-tails model instead of the lognormal in BSM one would get less favorable odds at the BSM 1-sigma and that might cause one to increase the safety margin by going further OTM maybe to the point of the call expiring worthless? A plot of GOOGL prices indicated the price distribution was not lognormal but fat-tails. Appreciate your comments.
If you thought Google would have rallied 100 points you shouldn't have put the trade on. If you didn't then in this case you were wrong. A cattail model won't change that fundamental decision.
What would have happened if you had hedged the deep OTM call with an slightly OTM call so that you were approximately delta and gamma neutral? Or, what if you had bought GOOG and sold those deep OTM calls to supplant income? My point is, imo, in equities, it makes very little sense to sell OTM calls unless you own the underlying (which is then a synthetic put). On the put side it makes sense because there is usually a steep smirk and at least you are selling expensive vega. But it flattens out for calls.