I've read a couple of times recently that institutions coming on board is going to make the price less volatile.
Since none of them are actually participating in the market they can not affect the price either. It is something similar to all the OTC trades that never had any meaningful effects on bitcoin price. In fact these institutional investors are also OTC trading instead of going on an exchange and placing buy/sell orders which means they can not affect the price. Not to mention that after they buy bitcoin they go away and create their own centralized market where they customers buy bitcoin from them and receive IOUs again not affecting anything about bitcoin.
-Institution buys $1bn worth of Bitcoin.
-Bitcoin price jumps up.
One time purchace, one time jump!
-Institution holds that Bitcoin essentially taking it off the market.
less than 20000 ($1 billion) bitcoin taken off the market is not a big deal!
-Buyers' money now equates to smaller units of Bitcoin (e.g. 0.9 BTC instead of 1 BTC).
-Those smaller transactions affect the overall price less.
The amount of money and the size of the order books didn't change in your scenario hence the volatility doesn't change.