# CMS Valuation methods

Does anyone know the difference in the valuation of CMS-related products? For example there are different ways to price this, 1. static replication using European swaptions 2. Linear TSR Model 3. LMM Model

All of which require some form of annuity mapping function and a convexity adjustment to correct for the pricing under different measure. I'd like to understand about (e.g) the difference between (1) vs (3), for example what risks am i missing using (1) vs (3).

Thanks!

Let $$S_t$$ the swap-rate and $$A_t$$ the associated annuity. You said that the convexity adjustment requires an annuity mapping function. That kind of approach is equivalent to calculate the following term $$E^A\left[G(S_T)\right]$$ where $$G$$ is the mapping (smooth) function.

One way to calculate that term would be to use a second-order Taylor expansion, and the martigale property of $$S_t$$ :

$$E^A\left[G(S_T)\right] \approx E^A\left[G(S_0)+G'(S_0)(S_T-S_0)+\frac{1}{2}G''(S_0)(S_T-S_0)^2\right]=G(S_0)+\frac{1}{2}G''(S_0)var(S_T)$$

To calculate $$var(S_T)$$, we need a model for $$S_t$$. A natural model would be a normal (Bachelier) or lognormal(Black) model. The problem of these choices is that we have to pick one volatility, in other words, the convexity adjustment will be consistent with only one swaption price( assuming the market is neither lognormal nor normal). It requires then to use a more complex model, such as LMM(a good one ,flexible enough to match the market cube) or SABR. The problem of using these complex models are their calibration, and the computation of $$var(S_T)$$ that can be heavy. It is worth noting, that one could have used the LMM without using any mapping function, but we would face the same issues.

Another method would be to use the Taylor expansion with integral remainder,

$$E^A\left[G(S_T)\right] = E^A\left[G(S_0)+G'(S_0)(S_T-S_0)+\int_{S_0}^{S_T}G''(x)(S_T-x)dx\right]$$

One can rewrite $$\int_{S_0}^{S_T}G''(x)(S_T-x)dx=\int_{S_0}^{+\infty}G''(x)(S_T-x)^+dx$$

Therefore,

$$E^A\left[G(S_T)\right] = G(S_0)+\int_{S_0}^{+\infty}G''(x)E^A\left[(S_T-x)^+\right]dx$$

Assuming that the annuity mapping function is exact, we know have an exact convexity adjustment, that only requires knowing all payer swaptions with strikes higher than the ATM forward rate. One will not need any simulation, but only prices. The drawbacks of that method are the calculation of the integral , and the need of admissible swaptions prices with very high strikes. Standard models such as SABR tends to overprice these products, and can have an impact on the CMS convexity terms.