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Derivatives of the Black-Scholes equation give us delta, IV, and other "greeks" for each individual options contract ( aka the equations to derive implied volatility for an option are very easy to come by ).

Question:
Q1: How do brokerages combine the IV readings of all the individual options in a chain for an expiration cycle, weigh them properly, account for things like Volatility skew, and come up with an overall implied volatility for the expiration that represents the Implied volatility of the entire chain?

Q2: Are there any common methodologies?

user3666197
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Drew Sturick
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If by "chain" you mean all the actively traded options of the same type across all strikes / maturities, then what you are asking is how banks / brokers fit an entire volatility surface to all the observed prices. Which accounts for both the vol smile and vol skew.

Generally you need to extend the black scholes model with additional degrees of freedom to produce the smile and / or skew behaviour commonly observed. One common model is the SABR model which has a nice closed form relationship to the black scholes IV and reproduces vol skew not vol smile (term structure), the extension called dynamic SABR model also produces vol smiles, there is also the Heston model which exhibits both vol skew and smile.

Jimmy
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This question is not suited for stack overflow in my opinion.

What most people refer to if they look at a single vol is the at-the-money (ATM) volatility. Sometimes also a VIX like measure. The VIX is the square root of the variance swap strike, computed via SPX options that are listed for trading on the Cboe.

Either way, the VIX and a 1m ATM IV is very similar as can be seen in this screenshot from a quant SE answer: enter image description here

Anyone interested in details about calculating IV can find plenty more details (some code for SVI, an interactive SABR chart and a general in depth explanation of what IV represents) in this answer on money SE.

AKdemy
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  • Usually one can vote to close it, provided one has enough reputation (instead of flagging it which does not have the option to close). It has not been closed, so the majority seems to disagree with my opinion. Also, it has several hundred views which shows that people look at it. I just added some resources that I think some might find useful. Again, some migth disagree. – AKdemy May 19 '23 at 21:01
  • IIRC, there is a "this question should be closed" option when flagging a question, which is not a vote to close, but it gets it into the close queue. And < 100 views/year is negligible. The low view count is probably why it hadn't been closed already. – miken32 May 19 '23 at 21:04