Why are manual market makers still predominant in options markets? Why haven't algorithms replaced these market makers, as they have for liquid stocks for example?
You need to differentiate between OTC and listed options in order to appreciate the fact market makers are still active and relevant in either segment:
Listed Options: Actually most listed options market making is governed by market making algorithms, however, most such algorithms are implemented with manual overlays. Something very similar goes on in the eFX world, where most quoting engines are fully systematized, however, spread tightening and widening can and is in fact overridden by humans who analyzed counter party profitability, trading actitivity such as frequency and such forth. This may not fully translate into listed options markets but the reason in general for listed options is that too much can go wrong when stocks, for instance, move by 10%+ within minutes post market-impacting events. Now that is true for stock markets as well but please keep in mind that most hft participants do not have an obligation to show prices, whereas options market makers are bound to stricter liquidity provision requirements, subject to each exchange on which such options are listed. So, while most all listed options contract market making is in some way or the other automated, there still are manual overrides at force.
OTC options markets: On the rates side, please appreciate that 90%+ of underlying notional traded goes through the OTC rather than listed markets. On the equities side that is somewhat lower but still significant, especially on the index side of things. With full customizability goes aboard the ability to fully automate. Now, everything in life can be automated, given a sufficient budget and no time constraints. But in a sense what chrisaycock pointed out is actually right: Car dealers, travel agents, and a host other middle men fight hard to stay in place and try to remain indispensable. Same with most OTC options markets: Fact is that which counterparty requests quotes, at what time, in what size, and especially the chatter surrounding an OTC trade is hugely important in brokering a deal, something that would be very expensive and time consuming to automate. For example, a keen understanding whether the requested size by a client is all that client is intending to trade or whether there is more that he/she possibly attempts to deal with other counterparties is part of the core job description of any trader, as it can hugely influence expected profit and loss for the market maker.
I am sure we will get there in the future, but for now most OTC market participants are not too interested in pushing into this direction simply because it also means they would cut their own fingers.