Overview
The long term bond is an asset that has a fixed interest rate that accrues over a long period of time. Typically, this asset is useful in pricing if the maturity of the long term bond can be chosen equal to the maturity of the derivative being priced. In such a case, the value of the bond today and at maturity is known, so that it effectively becomes a non-random variable and can be pulled out of the risk neutral expectation.
{% X(0) = \mathbb{E}_{\mathbb{Q}}[D(T)X(T)] = D(T) \mathbb{E}_{\mathbb{Q}}[X(T)] %}
Definition
Even though actual traded bonds are not quoted in continuous compounding terms, for the purposes of running the math, the rate used is the market rate quoted using continuous compounding. Because the bond has a fixed rate, the rate used is fixed and the value of the bond becomes.
{% B(t) = exp(rt) %}
This is the same as the
money market numeraire
where the rate is fixed. (except we use a long term rate instead of a short term rate)