# T-Forward Measure, LMM & the Zero T-bond

the zero-coupon T-bond is widely used in the industry as a tool to derive pricing formulas: for example it is used in the derivation of the Libor Market Model. The way in which it is often used implies that we assume a no-arbitrage relationship between the zero-coupon bonds and the Libor rates. As an example: if we assume discrete rates (as in the case of the Libor Market Model), we could write:

$$P(t_0,T):=\frac{1}{1+\delta L(t_0,t_0,T)}$$

Where $$P(t_0,T)$$ is today's price of the zero-coupon bond that pays 1 unit of currency at T, $$\delta$$ is the annual fraction corresponding to the Libor rate and $$L(t_0,t_0,T)$$ is the Libor rate between $$t_0$$ and $$T$$, observed at time $$t_0$$ (of course, from the above it follows that $$\delta=T-t_0$$).

My question: In reality, there is a very weak relationship between bonds and Libor rates. Even looking at the short-dated treasury bills in the US against the US Libors, the two instruments live their own, independent life. The above relationship certainly doesn't hold on most days, if on any day. The Libors are very stable, whilst the T-bills are quite volatile. Even the forward Libors implied by Eurodollar Futures or FRAs are quite independent of the T-bills. Doesn't this invalidate the usage of relationships such as the one stated above (and hence the entire LMM derivation)?

PS: I haven't even even mentioned non-US markets, where the relationship would be even weaker (because some governments don't even issue short-dated discount securities, and the ones that do certainly don't do it as often as the US). I would say that in most G10 markets, the assumption that we can price Libor rates from bonds is simply plain wrong.