A. Options Market makers are only obliged to quote (post two-sided quotes) 60% of the time, and then only $5 wide (with exceptions).
B. Furthermore, exchanges will generally only list options for securities where there is interest from a market maker.
- penny stock that has scant volume
It is unlikely an exchange would list options in a penny stock, nor would a market maker be willing to make markets in such a stock.
- very Out The Money option that has scant volume and open interest
This is common, for example the day after expiration, new options are listed in new series for new expiration months - the market maker will post quotes with no open interest and no volume. The market maker will calculate what the options are worth, add their desired spread (as long as it is equal to or less than $5 wide) and post their quotes.
- Must MM offer to buy 2 and 3?
If they are appointed as a Designated or Primary Market maker in that class by the exchange, then they would likely have to quote 99% of the series or 99% of the time, so most likely they would quote that series. However if they were only a Competitive Market Maker, they would only likely have to quote 60% of the time, so maybe not. However market choose which names they wish to be appointed, so may not choose to be appointed in a penny stock, and they can drop or sign up to new appointments from one day to the next.
- Won't the MM lose money on 2 and 3?
They decide the price and the spread. The spreads are also further adjusted by the spreads of the underlying security, for example if the underlying is $10 wide, the market maker might widen their $5 wide quotes by another $10. But if there is insufficient volume for them to delta hedge then they would likely drop the appointment or not quote and pay the fine.
- In general, if MM must buy or sell securities that it fore-knows will lose money, won't the MM lose lots of money?
The spreads allow them to cover any potential losses, and they can resign appointments. Ultimately, market makers who can't make a profit go out of business.