interest rates - 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

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