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If an underlying has a large bid-ask spread, does it mean that its options will have large bid-ask spreads too? Is there any relation between the bid-ask spreads of options and the bid-ask spreads of their underlying? If so, how and why?

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There is a relationship and it comes about two seperate ways. A live price of an option (so one traded without delta) will be wider in an illiquid market then a liquid one as the market maker would have to buy the delta hedge and cross that wide underlying spread. That is the easy case to see, however remember that an options value is in some sense determined by the cost of replication. So an underlying with a wide spread would be more costly to delta hedge as that spread would have to be crossed everytime the portfolio required rebalancing and thus would also increase the spread of options with delta.

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