Why the zero-volatility spread will be zero for an on-the-run Treasury Bond?
If you built the curve from strips, the z-spread would be negative.
I think on the run treasury bonds are valued using the arbitrage free valuation and use the spot rates for treasury. Z spread is the spread that when added to spot rates will make the present value of cash flows of non treasury bond equal to its price. Now in this situation we have the on-the-run treasury that is already using hte spot rates, so we do not have to add anything and that makes the z spread 0.
The topic is really more like a LIII topic, but that “on-the-run” piece is important and how you create that spot rate curve is important. First, you can’t really create the spot rate curve with OTR bonds because you have 4 pts after 1 year. That means you have to create the curve using STRIPS, all bonds, or off-the-run bonds. OTR bonds are always pricier than off-the-run bonds because they are in demand by the repo market as bond dealers short them to hedge interest rate exposure. That means if you created the spot yield curve by using any other bonds the Z-spread for OTR bonds would be negative. The “arbitrage-free” valuation runs into a billion problems of reconstituting an OTR bond from STRIPS which you just can’t do.