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I keep hearing things like \begin{align*} ``\text{Traders make their book delta-neutral at the end of each trading day''} \end{align*} I am wondering what this means, and why this is supposed to give the traders some peace of mind.

More specifically: How do the traders compute delta and decide how to hedge an option position? Do they assume the Black-Scholes model (or some other model) and then compute delta? In that case, what about model error (delta is the partial derivative w.r.t. price and is model-specific)? Thus, the delta they compute may be completely wrong!

Similarly, I have some market data from a financial data vendor, containing S&500 call and put option prices, as well as the delta, gamma, etc. of the options. In general situations like that, how are those quantities computed? Is it simply under Black-Scholes assumptions?

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Yes, you are right. There might be model errors. It depends on experience, personal choice and the mandate of the desk which model they use.

Using the model you use, you calculate your delta and hedge. You of hedge your delta but you are exposed to higher order greeks which would give you a delta exposure when the market moves.

As far market data goes, you should check the documentation. But most likely it would be black-scholes.

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