European options buyers cannot exercise early, so if the sellers intend to hold their contracts till expiration, then the volatilities in between do not matter. But that holds true for the buyers too. If they intend to hold the right till expiration, the volatilities in between do not matter either. But, both buyers and sellers do not have to hold their positions till expiration, they can both exit early and when doing that volatility matters.
Sorry, I'm just not sure what the point is. I'm trying but it's over my head. Here is what I know. Option seller always have more risk than option buyers with American style options. Option sellers don't have more risk with European style options. In fact option sellers have an financing edge in a PM account. The biggest risk is with cash settled, American options for the seller, as they can lose their hedge at anytime. Vol assumptions have nothing to do with it.
The important fact is this: the options seller wants to keep the credit, so usually he hopes to hold the option until it expires, because early closing a short option is never a good idea because one would lose much. Since early exercise and assignment isn't possible with European Style options, then this is a benefit for the seller, because he is free of the impact of what volatility changes otherwise would play if he would be early assigned. So, IMO a major/remarkable/important benefit of European Style options for the seller.
Not with equity options that are American Style. I lock in my profit by being assigned. There are two exceptions. Options on hard to borrow stocks and ITM options if the dividend is high enough that being assigned I lose my hedge and the dividend.
Hmm. being early assigned usually means losing all or part of the credit, or even more. Assignment happens because the counterparty (ie. the option buyer) makes a profit (meaning you make a loss), otherwise he wouldn't exercise...
Let me say it again... If you mark-to-market, volatility matters, regardless of the style of the options and whether one is a buyer or seller. Period.
Mark-to-market is not done usually by oneself, but by the broker etc. It has IMO no impact on the position itself. It is just an informal value for some bookkeepers (for margin calcs etc).