# 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?

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.
• 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. – user39039 Feb 28 '18 at 13:44
• 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 $t$. – user39039 Feb 28 '18 at 14:38
• 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. – Antoine Conze Feb 28 '18 at 15:09
• 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 ". – dm63 Mar 1 '18 at 4:48