Replicating CMS Rate Payoffs with Swaptions
Summary
The document explains how a constant maturity swap (CMS) rate payoff differs from the payoff of a vanilla swap. A vanilla swap's value depends on both the rate difference and the swap's DV01, which itself varies nonlinearly with the underlying rate. A CMS payoff is presented as linear in that rate, so simply holding a vanilla swap does not reproduce it.
The proposed replication starts with payer and receiver swaptions at the target strike, then adds payer swaptions at higher strikes and receiver swaptions at lower strikes. Their notionals are weighted to capture the payoff shape and the rate-dependent swap value. The document says the weights depend on the rate level, strike spacing, and number of swaptions. It describes this as computationally intensive, while noting that it can account for the swaption smile when estimating the CMS convexity adjustment. It points to Hagan's work as a reference, but provides no derivation, numerical example, or discussion of practical calibration choices.
Key ideas
- A vanilla swap payoff includes a DV01 factor that varies with the underlying swap rate.
- A CMS rate payoff is linear in the swap rate, unlike the vanilla swap payoff.
- A replication can combine payer and receiver swaptions at the target strike and at strikes on either side.
- The additional swaptions require calculated notional weights that depend on the rate level and strike grid.
- The method can incorporate swaption smile effects but is computationally demanding.
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Full text
# What is the replicating portfolio of swaptions for a constant maturity swap (CMS)? # What is the replicating portfolio of swaptions for a constant maturity swap (CMS)? How do you replicate the payoff of a constant maturity swap rate? That is, if the payoff of a contract pays the 5-year swap rate every year for 10 years, how would you replicate this payoff using swaptions? ## Answer by TheBridge (score 4) https://quant.stackexchange.com/a/2410 A good place to start is Hagan's paper Convexity Conundrum ...available on the web. ## Answer by user35980 (score 0) https://quant.stackexchange.com/a/70534 This is an important question and while Hagan's paper is the primary reference, actually understanding it can be a somewhat involved process. A simple and intuitive answer is not always easy to find. Here's an attempt. The payoff of a vanilla swap with (pay) fixed rate $K$ is $(S-K)*dv01$. The payoff of a CMS is just $(S-K)$. The level of the swap i.e. the $dv01$ is a non-linear function of the underlying rate $S$. Thus the CMS is a linear instrument (bright red line below) and the vanilla swap is non-linear (dark red line) in $S$. The CMS payoff can be replicated by: - replicate the original vanilla swap with long a strike $K$ payer swaption and short a strike $K$ receiver swaption. - make a portfolio of long vanilla payer swaptions and with strikes $K+i\epsilon$ where $\epsilon$ is a constant (say 50bps) and $i=1,...,n$ are the number of swaptions chosen, and appropriately weighted notionals - make a portfolio of long vanilla receiver swaptions and with strikes $K-i\epsilon$ where $\epsilon$ is a constant (say 50bps) and $i=1,...,n$ are the number of swaptions chosen, and appropriately weighted notionals The weights of the swaption notionals can be precisely calculated and depend on the level, $\epsilon$ and $n$. This approach, though compute intensive, is the most accurate way of calculating the CMS convexity adjustment and also has the added benefit of incorporating the swaption smile into the calculation.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.