Clearing houses cover any shortage of money due to one party not paying up. So if someone sells an option on an exchange, then doesn't pony up, the clearing house must. And if too many people don't pony up, the clearing house is in trouble.
Except it is far more complicated. Does Robin Hood allow the selling options? That would seem ridiculously stupid if it does, but let's say yes. The clearing house has nothing to do with the seller, or in fact probably with Robin Hood. It will be dealing with Robin Hood's dealer, probably a big bank (think JP Morgan or the like), and there's a cascade of who-owes-who.
Robin Hood clients ought to pay to Robin Hood if they owe money on short options, but if they fail, then Robin Hood must cover that to the broker. The broker wants to make sure it doesn't lose any money on trading with Robin Hood, because if it does, it still needs to make good to the exchange. And only if the broker fails, then the clearing house gets involved.
If you can only buy options on Robin Hood (or, in any case, cannot be short), then there isn't much of an issue there. The issue would arise if the broker cannot pay the option payouts, but that's very unlikely. Banks deal with option trading all the time, and spend lots of money on hedging their exposure to the underlying stock. This is essentially a solved problem, and the scale of what's going on is way too small to stress that.
Except it is far more complicated. Does Robin Hood allow the selling options? That would seem ridiculously stupid if it does, but let's say yes. The clearing house has nothing to do with the seller, or in fact probably with Robin Hood. It will be dealing with Robin Hood's dealer, probably a big bank (think JP Morgan or the like), and there's a cascade of who-owes-who.
Robin Hood clients ought to pay to Robin Hood if they owe money on short options, but if they fail, then Robin Hood must cover that to the broker. The broker wants to make sure it doesn't lose any money on trading with Robin Hood, because if it does, it still needs to make good to the exchange. And only if the broker fails, then the clearing house gets involved.
If you can only buy options on Robin Hood (or, in any case, cannot be short), then there isn't much of an issue there. The issue would arise if the broker cannot pay the option payouts, but that's very unlikely. Banks deal with option trading all the time, and spend lots of money on hedging their exposure to the underlying stock. This is essentially a solved problem, and the scale of what's going on is way too small to stress that.