When interest rates rise, bonds that are putable will go be put.

When interest rates fall, bonds that are callable will be called.

When we say volatility… we are saying interest rates moving, not specifically up or down ? right?

But when interest rates rise, will a bond with a putable option be worth more (compared to a bond without this option), as you get closer to the price that it can be put.

The same for a call option. When interest rates fall, will a bond with a callable option be more worth (compared to a bond without this option) as you get closer to the price that it can be called?

Owning a callable bond is equivalent to a long position in an option-free bond and a short position in the call option. As the option value increases, the value of the callable bond decreases (compared to the price of a noncallable bond).

I think im just confused when i see volatility in some questions, and interest rates up/down. Which are the same thing, but different answers, if the question specifically tells you where the volatility is headed… right?

If you know which way interest rates are headed, you’ll know whether the option will be in the money or out of the money: the intrinsic value of the option. If you only know that interest rate volatility is increasing or decreasing, you’ll know whether the time value of the option increases or decreases, but not the intrinsic value.

Not always,But yes If the conditions are profitable for issuer or investors they will go for it.

If the interest rate increases that means bond prices will fall down, And the putable bonds can be sold back at higher prices(If they feel this is the right time to sell and it is profitable). And for call option it is a vice-versa.

If interest rates decrease, a callable bond will sell at a _ lower _ price than an option-free bond, not at a higher price. The bondholder is short the call option (the bond issuer owns the option).

Suppose that you own a 6% coupon, €1,000 par, semiannual pay bond with 10 years to maturity; your buddy owns a 6% coupon, €1,000 par, semiannual pay bond with 10 years to maturity, currently callable at 104. The YTM on these bonds is 5%. Your bond sells for €1,078, but your buddy’s bond cannot sell for more than €1,040 (would you pay €1,077 for a bond that you could be forced to sell to the issuer at €1,040?).

Understand this in terms of option value and bond value. In callable, you are short the call because the issuer is long it. If interest rate decreases, the bond price will increase, which will give an incentive to the issuer to call it back. So, for him the option value increases, which means for you call option value decreases. I hope you get the drift now. Anyway, thanks for this question. Now, I know that I have to analyze all those questions fundamentally which I don’t get in the first shot.