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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?

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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.

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  • $\begingroup$ Thanks Antoine for the response. What happens in case that rates are deterministic? I mean , simply stated I have the following expression $A(t) = \sum_{i=0}^{n} \alpha_i e^{r(T_i-t)}$ where $\alpha_i$ are the tenors. $\endgroup$ – user39039 Feb 28 '18 at 13:44
  • $\begingroup$ If discounting rates are deterministic then both measures are identical. But it would be strange to assume a model where swap rates are stochastic while discounting rates are deterministic. That's why the trick of working under the annuity measure is neat: under the annuity measure the only stochastic dynamics you need to specify is that of the swap rate. $\endgroup$ – Antoine Conze Feb 28 '18 at 13:52
  • $\begingroup$ Maybe I should reformulate my question as. What is the difference between evaluating $\mathbb{E}[\max\{(s_T-s_K,0\}]$ under the forward swap measure or risk-neutral measure $\mathbb{Q}$ given the dynamics under Black's model $dF_t = \sigma F_t dW_t$ where $F_t$ is the forward swap rate at time $t$. $\endgroup$ – user39039 Feb 28 '18 at 14:38
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    $\begingroup$ When you write $dF_t=\sigma F_t dW_t$ the drift is zero so you are assuming that $F_t$ is a martingale under the measure you are working on. However martingality is only true under the annuity measure, as follows from the relation $F_t=PV_t[\text{float leg}]/A(t)$ which shows that the forward swap rate is the value of a self financing asset (the floating leg) divided by the annuity, therefore a martingale under the annuity measure. Martingality of the forward swap rate does not hold under the risk neutral measure, unless discounting rates are deterministic and both measures are the same. $\endgroup$ – Antoine Conze Feb 28 '18 at 15:09
  • $\begingroup$ Also, $E[max(S_T-S_K,0)]$ is actually higher under $Q$ than under the annuity measure. The difference between the two is due to the" cms convexity adjustment ". $\endgroup$ – dm63 Mar 1 '18 at 4:48

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