Why CMS Swap Rates Need a Convexity Adjustment
Summary
The document explains why a constant maturity swap coupon linked to a long tenor swap rate cannot be valued simply by substituting the forward swap rate as the expected coupon. In the example, one leg pays a short term LIBOR rate and the other pays a ten year swap rate, with OIS discounting. The key structural point is that a CMS swaplet pays once at a fixing related to the long tenor rate, whereas a swap of that tenor involves a schedule of cash flows. Hedging the single CMS payment with the underlying swap can therefore leave a different profit or loss at payment.
The answer describes the convexity adjustment as the correction needed for this mismatch and says the effect can theoretically be hedged from inception with a portfolio of swaptions. It recommends further reading but gives no adjustment formula, numerical example, or calibration method. Its comments about multi curve discounting are explicitly an opinion and limited to the context described.
Key ideas
- A CMS payment is a single coupon linked to a long tenor swap rate.
- The forward swap rate need not equal the expected CMS coupon under the relevant pricing measure.
- A swap hedge can leave a payoff mismatch because its underlying cash flow schedule differs from the CMS payment.
- A portfolio of swaptions can theoretically hedge the convexity effect from trade inception.
- The note gives no formula or numerical procedure for calculating the adjustment.
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Full text
# What is the reason for the convexity adjustment when pricing a constant maturity swap (CMS)? # What is the reason for the convexity adjustment when pricing a constant maturity swap (CMS)? I'm trying to wrap my head around pricing a Constant Maturity Swap (CMS). Let's imagine the following deal: 6m LIBOR in one direction, 10y swap rate in the other. The discount curve is derived from OIS. Naively I would just price this by taking the difference between the present value of cash flows from the forward 6m LIBOR rates and the present value of the cash flows from the forward 10y swap rates. I assume the cash flow from the swap leg is: 10y swap rate * notional But apparently this is not right, as quoting from here "the expected swap rate $\not=$ the forward swap rate" and this is the origin of the famous convexity adjustment. But why does the expected rate not equal the forward rate and how might one compute the difference? ## Answer by TheBridge (score 13, accepted) https://quant.stackexchange.com/a/2229 CMS adjustments in single curve context can be roughly explained if you consider a CMS swaplet by the fact that there is a single payment at the CMS rate at a single date and not on the whole strip of the underlying CMS tenor schedule. So if you are trying to hedge a CMS swaplet with the corresponding swap of CMS tenor length (with correct naïve nominal adjustment) then you end at the payment date with a swap that has a P&L different from the coupon you have to pay that day. Theoretically, though, you can statically hedge this effect from the very beginning of the trade by using a "continuous" portfolio of swaptions. In this regard I strongly recommend that you read the excellent exposition of P. Hagan " Convexity Conundrums : Pricing CMS Swap, Caps and Floors" freely available on he web. Regarding the OIS - Libor discounting effect, it is (IMO) a second (or third) order effect and doesn't differ from the single curve framework in a perceptible way. Best regards
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