Recall that the OAS is a spread that adjusts for the difference between the modeled value of a bond (binomial tree modeling) and the bonds market price. It is an additional discount to break the modeled price down to market price.

For the same bond, the OAS will vary depending on the volatility assumption used. As the assumed level of volatility increases in a model, the OAS for a callable bond will decrease. For a putable bond, the opposite holds, increases in volatility will raise the OAS necessary.

This is because a higher level of volatility, in the case of a callable bond, increases the value of the call option, which decreases the price of the callable bond. Thus a lower OAS will be needed to match market prices. For a putable bond, volatility increases the value of the put option, which increases the price of the putable bond. The bond is now further from its market prices and will need a larger OAS spread.