Non-sequitur regarding putable options? 207, expression, regarding, increase

kickthatcfa

pg. 207 in the Schweser text:
“As such, it can be seen in the expression that the value of a putable bond will also increase as volatility increases.  Intuitively, this makes sense; investors are willing to pay more for a bond that gives them the right to sell it at a price greater than the market value.”
The second sentence doesn’t quite follow for me, because if so we are assuming volatility corresponds with a drop in price.  Why is that?  Shouldn’t there be volatility also if the price rises a lot?  I mean this goes with the VIX only going up when prices drop.  Could someone explain to me why we assume volatility only goes up when prices drop and not the other way around?  Or am I missing something?