# Bond yield: is it martingale with respect to risk-neutral probability measure of some numeraire?

Let $t$ mean current time, let $T_0, T_n$ mean two times such that $T_0\le T_n$, and let $y_t[T_0, T_n]$ mean the forward swap rate of a swap starting at $T_0$ and ending at $T_n$. (I am ignoring $T_0+2$ issues, and assume that the swap starts at $T_0$.)

Then under the annuity numeraire $N_t = P_t[T_0, T_n]$, the forward swap rate $y_t[T_0, T_n]$ is a martingale under the risk-neutral measure associated with $N_t$. This follows from the fact that $y_t$ is a ratio of a portfolio of assets by $P_t[T_0, T_n]$. Indeed, $$y_t[T_0, T_n] = (Z(t,T_0)-Z(t,T_n))/P_t[T_0,T_n],$$ where $Z(t, T_i)$ means the zero-coupon bond from $t$ to $T_i$.

Is there a corresponding numeraire for the yield of a bond?

I am guessing the answer is no under some mild assumptions because bonds are tradeable and their price has a non-zero second derivative with respect to yields, but cannot hack through the thicket of results at the moment.

Thanks in advance, any help appreciated!

• Note that, a yield can be defined in many ways, and some use the yield as the coupon rate. If you can define the yield specifically, people may be able to explore from there. – Gordon Aug 7 '15 at 18:22