OAS and level of interest rate volatility

If the OAS of a straight bond is 48bps, putable bond and callable bond with similar liquidity risk and credit risk compare to the straight bond should also have a OAS of 48bps. Is this correct?

If the above is true, then why during high level of interest rate volatility putable bonds should have a higher OAS? (I understand OAS is a measure of the attractiveness of the bond, and putable bonds protect investors during times of volatility)

I am having trouble bridging the two concepts.