# 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?

A good place to start is Hagan's paper Convexity Conundrum ...available on the web.

• @kmcoy : beside my answer I think that you should ask more precise questions once you have read this paper where the static no arbitrage hedging procedure is clearly exposed. Best regards Commented Nov 21, 2011 at 8:44
• The link is apparently broken Commented Nov 29, 2020 at 15:28
• you can find it here :researchgate.net/publication/… Commented Nov 29, 2020 at 19:38

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:

1. replicate the original vanilla swap with long a strike $$K$$ payer swaption and short a strike $$K$$ receiver swaption.
2. 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
3. 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.