Is there a way to understand why that when interest rate volatility rises that OAS for a callable bond drop drops and OAS increases for a putable bond? Would this be the correct way of interpreting it? Z-Spread =OAS + Option Cost (Putable Bond) Z-Spread =OAS - Option Cost (Callable Bond)
Callable bonds: As volatility rises, the option value rises, and the callable bond value falls. As it falls, it gets closer to its straight price, hence you do not need a very high OAS to adjust the callable bond value to its straight price any more.
Other way around for putable bonds.
Thanks. It is actually easier to understand it that way, “it gets closer to its straight price” but Schweser says that when interest rate volatility increases, “a callable bond will be lower, and therefore closer to its actual market price”. Not sure whether the “actual market price” refers to the market price of a straight bond, or of a callable bond?
Why when interest rate volatility increases, the OAS of a putable bond is higher? 
Neither is anyone else; the wording’s lousy.
Take a look at the link I posted above.
In short, higher volatility means lower prices on the bottom nodes of the tree, meaning more likelihood of the bond being put, meaning higher average cash flows in the tree, meaning that a higher discount rate is needed to get to the same market price, meaning that the OAS is higher.