Explain the effect of interest rate volatility on what a callable or putable bond is worth

Option value rises with volatility, whoever owns the option. A callable bond equals a straight bond minus a call held by the issuer, so higher interest rate volatility raises the call's value and lowers the callable bond's price. A putable bond equals a straight bond plus a put held by the investor, so higher volatility raises the putable bond's price. The straight bond's value on a calibrated tree does not depend on volatility. The same reasoning covers capped floaters, which lose value as volatility rises, and floored floaters, which gain. A common slip is to get the direction for callables wrong by forgetting that the bondholder has sold the option.