Enjoy your hat! The one you brought up, but aren't following the context.

I don't have the supplies, it's your hat buddy! Wear it with pride.

You are already wearing one, you just don't realize it yet........

That is incorrect, although your point is fair. You do need to make some adjustments for dealing with American options. The theoretical framework around BS is for European options, and doesn't directly consider early assignment. The model is also built on a normal distribution curve, and the stocks don't behave as a normal distribution curve. It is a model, it certainly isn't "perfect", but it is a really good model that help to understand the mechanics behind option valuations.

If you are going to put together a true "trading" program, that is the way to go to skin the cat correctly. Computer aided drawing on graphs isn't going to cut it, not by a long shot.

For options, you should study up and understand the Black-Schoels model. Best if you can write out the derivation yourself. Then look at the market empirically, where the results in the real world differ from the how the market treats those aspects. Read up on the Greeks, so you can understand them. Although depending on your approach certain Greeks may be totally unimportant to you. Understanding that is a lot less difficult than putting together a trading program.

Probably books on Riemannian geometry, differential geometry, and complex geometry. You might just want to look at the math textbooks for the PhD program that James Simons went through, that is probably a good starting point.

The stock market isn't normally distributed, so using normal distribution curves for your IV is problematic.