When a client sells options - broker keeps the margin money?

Larger clients normally pledge securities to trade, don’t leave free cash. Also, the yield on the float income is much lesser as the money gets used for a bunch of things (including quarterly settlement when funds are sent out). But yes, float income is revenue for all financial services firms (including folks who run wallets - paytm, mobikwik, etc). It usually contributes 15% of revenue for brokers. FYI last year Interactive brokers made a billion dollars in revenue just from the float income in the US.

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