Why Swaption Pricing Uses the Annuity Measure

# Change of Numeraire to price European swaptions


# Change of Numeraire to price European swaptions












In the pricing of a European swaption, it is common to use the annuity factor $A(t)$ as the Numeraire. I was trying to write down the pricing formula via the bank account as numeraire to see if they were equivalent. Let $C(t)$ be the price at time $t$ of a European swaption with strike rate $s_K$ and swap rate $s_T$ that prevails at maturity $T$ then the price can be determined via martingale pricing:

(1) Bank account numeraire: $P(0,T)$ under the risk-neutral measure $\mathbb{Q}$: \begin{align} C(0) &= P(0,T) \mathbb{E}_{\mathbb{Q}}\left[\frac{A(T)\max\{s_T-s_K,0\}}{P(T,T)}\right] \\ &=P(0,T)A(T)\mathbb{E}_{\mathbb{Q}}[\max\{s_T-s_K,0\}] \\ &=A(0)\mathbb{E}_{\mathbb{Q}}[\max\{s_T-s_K,0\}] \end{align} Assuming that the swap rate $s_T$ follows a lognormal process then the above expression essentially comes down to evaluating a call struck on $s_K$ under the risk-neutral measure.

(2) Annuity as numeraire: $A(t)$ under forward swap measure $\mathbb{A}$: \begin{align} C(0) &= A(0) \mathbb{E}_{\mathbb{A}}\left[\frac{A(T)\max\{s_T-s_K,0\}}{A(T)}\right] \\ &=A(0)\mathbb{E}_{\mathbb{A}}[\max\{s_T-s_K,0\}] \end{align}

Is the above correct? I am much more used to value under the risk-neutral measure, so my question is that it seems that taking the expectation under the risk-neutral measure or the forward swap measure should give the same result. Why is this so? Is there some way by using Girsanov's theorem to prove this?

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/38531

Your (1) is incorrect because the annuity $A(T)$ is stochastic (it depends on discount rates on expiry) and therefore cannot be taken out of the expectation $E_Q[]$. This is why one resorts to pricing under the annuity measure.