So, someone needs to start a total anonymous identityless exchange.
People can't even get their funds back from exchange owners whose identities are known. What makes you think that anyone who sets up a totally anonymous exchange is going to be trustworthy?
Another distributed blockchain that holds the information about shares and there would be no need to trust anyone. The exchange pages would hold no BTC or shares, only information about prices and sell/buy orders, which it would have extracted from the blockchain. Actually, the whole trade could be made without any central exchange. Just a client (like bitcoin-qt) which would do everything and store the information on a blockchain.
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