system
Given the following post, respond as if you're commenting on Reddit:
human
Why do we need advanced options pricing models if we have the Black-Scholes formula?

Specifically, where is there emphasis on the development of robust options pricing models (finite difference, finite element, etc.) for the Black-Scholes PDE if we have the Black Scholes formula that we can use to evaluate the value of an option for any time, volatility, etc.

For context, I am a senior applied math undergraduate student in the US. I am not *super* familiar with a lot of econ/finance topics.
gpt
The reason is simple - Black-Scholes is simply a popular *attempt* to model option value and does not, by definition, reflect actual option value. In practice, different assumptions about the underlying are often false and this is particularly evident over longer periods of time. 

Naturally, the *actual* value of an option depends on the subjective probabilities of the individual market participant and their notions about the future price of the underlying over a given period of time. This naturally leads into probability theory which, to complicate matters, has a non-mathematical "logical of probability" branch.