So a straight bond is a bond without any options. A bond without any options is invulnerable to interest rate volatility.

Higher volatility = higher call price = lower callable bond price as investors are short the call option. On putable bonds, the effect is the opposite as investors are long the put option. So if volatility goes up both options will go up in value but the callable bond will fall and putable bond will rise in value. Rising interest rates will lower the call price as the issuer is less likely to exercise it due to lower bond prices. The effect on put options is the opposite as higher interest rate = higher uncertainty of cash flow = more likely to be exercised by the investor.

What it means is that, for the moment, interest rates are the same at all maturities.

But they may not be the same at all maturities next month. Or next week. Or tomorrow. Or an hour from now.

The volatility of interest rates affects the value of embedded options, but it doesn’t affect the value of the underlying, option-free bond.

Rising or falling interest rates affect the value of embedded options as well as the value of the underlying, option-free bond.

Think about how you compute a straight bond’s value: you discount the cash flows with current interest rates. There is no volatility term in the formula, so volatility doesn’t affect the value.