# CMS Pricing - Convexity Adjustment by Replication [closed]

I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex. I'd be glad if you can provide me with simpler articles or spreadsheets to give an idea about replication of swaptions.

## 1 Answer

The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate. Let us start with the fair value of a swaption under the annuity measure $$\mathcal{A}$$ with tenor at time $$\tau$$: $$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]$$ Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $$T$$. We are, thus, in a change of measure from the annuity to a $$T$$-forward measure. Denoting the bond value value today maturing at $$T$$ by $$B_{t,T}$$, the CMS flow at time $$t under the annuity measure is $$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t\left[\frac{\mathcal{S}(\tau)}{B_{\tau,T}\mathcal{A}(\tau)}\right]$$ whilst under the $$T$$-forward measure it is: $$\mathbb{E}_t^T[\mathcal{S}(\tau)]=\mathbb{E}^\mathcal{A}_t\left[\mathcal{S}(\tau)\frac{dT}{d\mathcal{A}}\right]$$ with the Radon-Nykodim derivative $$\frac{dT}{d\mathcal{A}}=\frac{\mathcal{A}(t)B_{t,T}}{B_{\tau,T}\mathcal{A}(\tau)}$$. The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $$\mathbb{E}^S_t[1/B_{S,T}|\mathcal{S}_\tau]$$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.