[luca3v]

You can see that the market deviates from the model and accounts for the fact that the Brownian motion model of Black-Scholes underestimates the probability of big moves: typically, options for the same security and the same expiration date have different IV, with options ATM having a lower IV and options deep ITM or deem OTM having larger IV.

If the market believed in the model, options for the same security and the same expiration date would all have the same IV, which would be whatever volatility the market thinks the security is going to have.

[smabie]

You can just input diff IVs for BS for diff strikes, you don't have to actually ditch BS.

After all vol is the free parameter for BS

That being said, not an expert on non-d1 products so I could be wrong about how this is dealt with in practice