CMS Convexity Adjustment with Taylor Expansion and Swaption Replication
Summary
The document compares ways to value constant-maturity swap (CMS) products, focusing on how an annuity mapping function and convexity adjustment account for the change of measure. One approach approximates the expected mapped swap rate with a second-order Taylor expansion. This requires modeling the swap rate’s variance; a single normal or lognormal volatility may fit only one swaption price, while a more flexible model such as LMM or SABR can better match a market volatility surface, at added calibration and computational cost.
A second approach uses the Taylor expansion’s integral remainder to express the adjustment as an integral over payer swaption prices at strikes above the forward rate. Under an exact mapping assumption, this gives an exact adjustment without simulation. Its practical limits include evaluating the integral and obtaining reliable prices at very high strikes; the response notes that standard models such as SABR may overprice those swaptions. The discussion is conceptual and does not provide a numerical comparison of model risks or calibration results.
Key ideas
- A second-order Taylor approximation reduces the CMS mapping expectation to a term involving swap-rate variance.
- A normal or lognormal volatility choice may be consistent with only one swaption price.
- Flexible models such as LMM or SABR can fit a broader volatility cube but may require costly calibration and variance computation.
- An integral representation can calculate the convexity adjustment from payer swaption prices above the forward rate.
- The integral method depends on an exact mapping assumption and reliable high-strike swaption prices.
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# CMS Valuation methods
# 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!
## Answer by Canardini (score 2)
https://quant.stackexchange.com/a/50047
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.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.