Can somone please explain why as volatility increases, OAS on a callable bond decreases and on a putable bond OAS increases?

Hi,

it is the *assumed* volatility in your model.

First you need to know, the price of the bond is stale, lets say 98. You calibrate your model so that you derive at this price.

If you now increase the volatility in your model, the Option is worth more, and makes the bond cheaper (as it is bond - option) To offset this you need to reduce the OAS in your model to get again the price of 98.

For putable, lets say the price is 103. If you increase the volatility in your model, the price would increase (as it is bond + option). To offset you need to increase your OAS to get again the price of 103.