Riiiight... Your PNL is based on the mark-to-mkt valuations of your options positions. These valuations are sensitive to levels of implied vol, to a lesser or greater degree. Again, think about the logic you have applied to determine that "development of volatility plays a role for the option buyer". Things cannot possibly be any different for the opposite position.
No, Mister! Please read again: what I said is specific to selling of options of the European Style only... Do you understand now, and do you see the difference?
No not really. The ability to exercise early provides a different value for the options. In fact, it tend to make it worth less because you have lost a right to add value by having the ability to early exercise it when it is to your benefit. However, it does not effect volatility. It is possible to have an ITM option worth less than parity because of cost of carry, when that does not happen with american options.
Huh? What difference does it make whether the option is European or American? The development of volatility during the life of the option matters to sellers and buyers alike, regardless of the style of said option.
Hmm. not sure I get your point. I'm recapitulating: Early exercise by the options buyer leads to early assignment of the options seller. This (early exercise and assignment) is possible with American Style options only, not with European Style options. Then I think the American Style options should be worth more than European Style options (hmm... questionable; maybe both should be worth the same) But the question was whether volatility plays any role for the seller of an European Style option _after_ opening the position and upto the expiration date. I would say volatility plays not a single role, so there is no volatility risk with selling of European Style options.
Perhaps I'm not understanding the questions as you are phrasing it. I'm telling you that American vs European only effects the value calculations of ITM options. It does not effect anything else that is material. If you would like a different response, you will have to ask the question in a different way.
As in the OP of mine stated: does the European Style options seller need to take care of the (future) development of the volatility (IV and/or HV of the underlying)? My point is: no. And this is IMO an important difference between the American Style and the European Style options for the seller. So, with European Style options there is no volatility risk (for the seller) if one intends to hold the option until expiration. IMO this is a valuable knowledge for the seller for risk eliminating... Meaning: one should prefer to sell European Style options instead of the American Style options, as the latter has the risk of volatility changes... Hmm. IMO volatility changes play for all options (regardless of its moneyness) a big role during the life-time, except for sold European Style options if one intends to hold it until expiration, what sellers usually are after.
The implication of the volatility change is different for the seller of European Style options. Don't you think so? Such a seller does not have any volatility risk whereas sellers of American Style options and buyers of any style options have the risk of volatility changes during the lifetime of the option. The point is: the European Style option seller does not need to take care of the volatility after opening the position (assuming he will keep the position until expiration).
When you say," The implication of the volatility change is different for the seller of European Style options.", I'm not sure what you mean by "volatility change".
Doesn't really matter whether it be a change in IV and/or in HV. Example: Code: A 3-month option of HV=30%: Spot=100.00 Strike=100.00 ExpDays=63 Interest=0.0000% Dividend=0.0000% Vola=30.00% --> Call=6.00 Put=6.00 Cdelta=0.529893 Pdelta=-0.470107 gamma=0.026521 After 1 month HV rises to 40%: Spot=100.00 Strike=100.00 ExpDays=42 Interest=0.0000% Dividend=0.0000% Vola=40.00% --> Call=6.50 Put=6.50 Cdelta=0.532537 Pdelta=-0.467463 gamma=0.024349 If HV had stayed at 30% it would be worth this: Spot=100.00 Strike=100.00 ExpDays=42 Interest=0.0000% Dividend=0.0000% Vola=30.00% --> Call=4.90 Put=4.90 Cdelta=0.524415 Pdelta=-0.475585 gamma=0.032512 All such changes would not affect the seller of European Style option, but would possibly cause concerns (bad or good) for all other option sellers, and all option buyers of any style.