Long callable bond = long option-free bond + short receiver swaption

Question

I have just read that long callable bond = long option-free bond + short receiver swaption

I don't see how the above equivalency works. In my head:

If interest rates are going down, a long callable bond is disadvantageous to the holder,
because it would be called and the company would refinance at lower rates.
 	A long bond would be worth having as your coupon would be higher (likely)
than current market rates.
 	You would also want to be long rather than selling a receiver swaption....

So I get how a short receiver swaption and long callable bond are equivalent, but I don't see how a long option free bond is? I'd be grateful if someone could explain this! Thanks :)

Sophie Macon · Answer

Let's run through a scenario where the interest rates drop:

Long callable bond: When interest rates drop, the bond holder (you) loses out. The bond issuer calls back the bond and reissues at lower interest rates.
 	Long option-free bond + short receiver swaption - when interest rates drop:

Long option-free bond: rises in price, so the bond holder (you) goes $$$ up, and you receive fixed interest payments.
 	Short receiver swaption: If you short a receiver swaption you're selling the option to receive fixed interest payments, so when interest rates drop, the option is exercised and you have to pay fixed and receive floating.

Your fixed payments from your long option-free bond therefore goes to pay your short receiver swaption, so you net receive at current variable interest rates, same as the long callable bond.

Does the rise in price of the long option-free bond mean that it's NOT equivalent? Nope, that gets cancelled out too.

The rise in price of the long option-free bond is due to your right to receive (higher) fixed interest payments when interest rates drop. This rise in value is negated by your liability to pay fixed due to your short receiver swaption. So price-wise, again it's a flush with the long callable bond.

Hope that makes sense? Let me know.