In Hagan's paper on valuing CMS swaps (Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors), there is:
So the swap rate must also be a Martingale, and
$$E \big[ R_s(\tau) \big| \mathcal{F}_0 \big]=R_s(0) = R_s^0$$
To complete the pricing, one now has to invoke a mathematical model (Black’s model, Heston’s model, the SABR model, . . . ) for how $R_s(\tau)$ is distributed around its mean value $R_s^0$. In Black’s model, for example, the swap rate is distributed according to
$$R_s(\tau) = R_s(0)e^{\sigma x\sqrt{\tau}-\frac{1}{2}\sigma^2\tau}$$
where x is a normal variable with mean zero and unit variance. One completes the pricing by integrating to calculate the expected value.
And he stopped here. So I am trying to "complete the pricing by integrating", but I am not sure of what he actually meant. I think he just wants to get the swap rate $R_s(0)$, but I am not sure.
This integration would lead to:
$$ R_s(0) = E \big[ R_s(\tau) \big| \mathcal{F}_0 \big] = \int_{-\infty}^{+\infty} R_s(\tau) \frac{-\frac{x^2}{2}}{\sqrt{2\pi}}dx = \int_{-\infty}^{+\infty} R_s(0)e^{\sigma\sqrt{\tau}x-\frac{1}{2}\sigma^2\tau} \frac{-\frac{x^2}{2}}{\sqrt{2\pi}}dx $$
This does not make sens, since $R_s(0)$ is on both side of the equation. Maybe I am not looking at it the right way, and he meant something else. Or maybe my integration is wrong.
I am a bit lost here. Any help would be appreciated to understand how to "complete this pricing".