# Is there a way to get convexity adjustements for any CMS-payoffs?

In the litterature we specify a dynamic for $$\frac{P(T,T_{p})}{A(T)} = G(S(T))$$ for each Swap rate S(T) , and there are supposed independant so that we can obtain some value using copulas for calculing the CMS spread with payoff $$( S_{1}(T)-S_{2}(T) - K)_{+}$$. But in reality the swap rates are not independant so that we cant'suppose the same G(S(.)).How do we account for this ? Is there a way to model those " annuity mapping function" consistent with swaps rates dependance?

To the extent that I understood your question:

1. You want to capture the marginals of the 2 rates perfectly, since they will be your hedging instruments. You can correlate them with a copula.

2. You need to model a change of measure from the annuity to the T-forward measure. This can be done in a simple hull white setting.

Pricing is then trivial.

Let me know if I misunderstood the question.

• Yes , that's pretty much it ,The distribution of $S_{1}(T)$ and $S_{2}(T)$ in their annuity measures can be mapped into the distribution under the $T_{p}$ forward measure using standard techniques like the copula method. The problem is not the copula method itself but rather the meaning of the parameters for the dependance structure. It should reflect both the correlation of rates observed at the same time and their inter temporal de-correlation.\\ What should i chose to account for this in the change of measure , which model better suits for this ? Libor Market model , hull white setting ? Jun 22 '20 at 8:41
• How would the copula method help in modeling the change of measure? Can you describe it? What you need to model is the annuity (or possibly bond/annuity) as a function of the swap rate. You can do this in a HW setting. This is sufficient to get the distribution to the Tp measure. The copula is to only correlate both the underlyings (i.e go from marginal distributions in the Tp measure to the joint). Jun 22 '20 at 10:45
• Ok thank you a lot , and if i have two different fixing dates is it the same thing ? or the distribution to the T-p measure will be modified ? Jun 24 '20 at 9:02
• Interesting. If you have significantly different fixing dates then the copula approach would not really go through. You will have to make assumptions about the delay effect in some way. You can still price in the Tp measure, but keep in mind that your product is starting to become more and more like a forward starting swaption, so you will nees stochastic vol if delay is significant. Jun 28 '20 at 2:58