CMS Convexity Adjustment from Swaption Replication
Summary
The document sketches why a constant-maturity swap (CMS) payment differs from a swap rate’s value under the swaption annuity measure. A swaption payoff is naturally valued under that measure, while a CMS coupon paid at a later date is expressed under the corresponding forward measure. The adjustment comes from this change of measure, whose density depends on the swap annuity and the bond maturing on the coupon date.
To calculate the adjustment, the method needs an approximation for the conditional expectation of the inverse bond price given the swap rate. The answer points to approaches that assume a non-stochastic Libor–OIS basis or express the relevant bond prices in terms of swap rates to obtain the swap-rate density. The post gives a conceptual outline and references a more detailed treatment, but it provides no derivation, numerical example, or comparison of the approximations; readers need further material to implement or assess the method.
Key ideas
- CMS convexity adjustment reflects a change from the swap annuity measure to a payment-date forward measure.
- The change-of-measure density involves the swap annuity and the bond price at the coupon date.
- Computing the adjustment requires modeling the conditional inverse bond price given the swap rate.
- Possible approximations include assuming a non-stochastic Libor–OIS basis or relating bond prices to swap rates.
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# CMS Pricing - Convexity Adjustment by Replication
# CMS Pricing - Convexity Adjustment by Replication
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.
## Answer by FunnyBuzer (score 7)
https://quant.stackexchange.com/a/43985
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 $t<T$ 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.