Why a Forward Swap Rate Is a Martingale Under Its Annuity Measure

Summary

The document explains that a forward swap rate is a martingale under the risk-neutral measure associated with the swap annuity numeraire. It presents the rate as a ratio: the difference between zero-coupon bond values at the swap’s start and end dates divided by the annuity value. This numeraire choice supplies the stated martingale property.

The question asks whether a similar numeraire can make a bond’s yield a martingale. It offers no resolution, derivation, or evidence for that question, and the proposed concern about the nonlinear relationship between bond prices and yields remains a conjecture. The useful point is the connection between choosing a numeraire and obtaining a martingale for a tradable ratio; the document does not establish a corresponding result for bond yields.

Key ideas

• A forward swap rate is presented as a ratio of a bond-value difference to the swap annuity value.
• Under the risk-neutral measure associated with that annuity numeraire, the forward swap rate is a martingale.
• The document asks whether bond yield has an analogous numeraire but does not answer it.
• The concern that price curvature with respect to yield prevents such a result is posed as a conjecture.

Tags

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


# 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!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.