Actually the bank does more than just issue coins and detect double spending. The bank is the ultimate authority on whether any coin is valid (just like with p

Actually the bank does more than just issue coins and detect double spending. The bank is the ultimate authority on whether any coin is valid (just like with paper money), in the sense that the bank can refuse to accept a deposit. This is crucial to the security definition of ecash -- crucial because the security definition of any cryptosystem that is trying to achieve "electronic money" must in some way define "money." Also, double-spending detection can be defined in several ways. My view is that the best definition includes the no-frame-up property and the "full tracing" property. A system that supports full tracing allows a cheating user to be blacklisted completely, since the double spending detection procedure produces a value that can be used to check if any coin was spent by the cheating user (basically, users have secret keys they use to spend; double-spending will reveal the secret key used for double spending). This is a nice property because it allows the bank to issue blacklists of cheating users, which are enforced by the bank's refusal to accept any coins that were spent by that user (even those not double spent), and thus no other party is willing to accept payments from blacklisted users since the coins are effectively worthless (as the bank will not accept them for deposit). In other words there is a meaningful way to punish a cheating user. Finally, there is no reason these systems must naturally be "stablecoins." The bank need not peg the coin's value to anything, it can float freely like any other currency and the bank can choose any monetary policy, including one that results in significant volatility. The bank could even emulate Bitcoin by setting a fixed limit on coins in circulation (the different between the number of withdrawals and number of deposits) and periodically releasing more coins (maybe giving them in fixed amounts to randomly chosen users to simulate mining). If the coins did have stable values or were pegged to a currency with stable value, that would only be because the bank itself thought it was a good idea (perhaps in response to customer demand), not because there is something inherent in ecash that promotes stability.