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Why SABR Can Misprice Constant Maturity Swaps

Article Quant Q&A · Author: Averroes
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Summary

The document explains why SABR volatility can be unreliable for pricing constant maturity swaps (CMS). CMS coupons are especially sensitive to the far right tail of the implied volatility surface, so pricing often depends on behavior at very high strikes. The discussion says that CMS market rates suggest implied volatility eventually flattens toward a constant, while SABR can imply a rapidly rising smile in that region.

The practical concern appears in replication with cash-settled swaptions: the model’s heavy right tail can make the calculation depend on unrealistically high strikes. The answers suggest that models built around the terminal swap-rate distribution may fit CMS behavior better than a stochastic volatility model whose wing fit is weak. These are concise qualitative explanations rather than a detailed derivation, calibration study, or comparison across market data; the document does not establish that every SABR specification or CMS trade will show the same degree of error.

Key ideas

  • CMS pricing is sensitive to the implied volatility surface at very high strikes.
  • SABR may produce a right tail that grows too quickly for CMS pricing.
  • Cash-settled swaption replication can require implausibly high strikes under the model.
  • Terminal swap-rate models may be better suited when pricing depends heavily on the terminal distribution.
  • The document offers qualitative guidance rather than empirical comparisons or a universal model ranking.

Tags

Fixed incomeVolatilityDerivatives pricing
Full text
# Why is the SABR volatility model not good at pricing a constant maturity swap (CMS)?


# Why is the SABR volatility model not good at pricing a constant maturity swap (CMS)?












I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?

## Answer by olaker (score 11)

https://quant.stackexchange.com/a/2050

Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that

> an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the option market in extreme strike regions and indicate that implied volatilities flatten out and converge asymptotically to a constant. This behavior is not consistent with the rapidly diverging asymptotics which are implied by SABR.

## Answer by Mark Joshi (score 7)

https://quant.stackexchange.com/a/2780

The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.

## Answer by user35980 (score 1)

https://quant.stackexchange.com/a/75372

Late to the game here and an already well-answered question, but an important one nonetheless. In a nutshell, stochastic volatility models are generally a weak solution for products that depend mainly on the terminal distribution (and CMS is a ubiquitous case of this) because of poor fitting of the vol surface (especially on wing strikes). Terminal swap rate models (linear, exponential ...etc) are usually a much better option.

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.