Hi,For a convex function, E(f(x))>= f(E(x))Now for a plain vanilla swap, E(f(x)) = f(E(x)), where f is Present value of swap and x is the stochatic swap curve. E is expectation under risk-neutral measure.i.e. we use the E(x) or the expected forward rate (under risk-neutrality) to be the realized forward rate.QUESTIONS:1. Could someone explain why? What is so "non-convex" (or linear) about the plain swap?2. Why is there convexity for a CMS swap (10y CMS rate paid every 3m)?\gracias

When you're pricing a regular swap (eg. quarterly payments indexed off 3M Libor), you have swap rates on the market which are quoted precisely for this instrument. You can thus price it directly from a discount curve built off the market swap rates. To answer question 2 take the following example: a semi-annual swap indexed off 6M Libor and a quarterly swap indexed off 6M Libor. In the first case we have quotes readily available on the market; in the second case we don't. When we compare both products we see that the second swap (quarterly indexed off 6M Libor) has a larger reinvestment risk that the first. For this extra risk investors need to be compensated in the form of a larger rate (this difference is called the convexity adjustment, which also exists between futures and forward rates).The main point is that whenever there is a mismatch between the term of the floating rate and the frequency of the swap we can't simply use market rates since these don't cater for this fact. The extra risk brought about by a long-maturity CMS rate paying frequent coupons must be accounted for by introducing a convexity adjustment.

So you're saying, convexity arises principally due to reinvestment risk? Could you elaborate more on the reinvestment risk for a 6mLIBOR paid 3monthly?How about a 3mLIBOR paid "in-arrears"? How about a 3mLIBOR paid every 6 months?thanks

The 6m Libor paid every quarter faces reinvestment risk when compared to the 6m libor paid semi-annually. In the second case, very time the rate refixes you have fixed the interest rate that will be applicable of the next 6 months. In the first case, after 3 months you will get a different refixing which may or may not be more advantageous. It is this uncertainty in the total interest rate which is applicable over each 6m period that means that the first case is more risky than the second.A 3m libor paid every 6M also needs a convexity adjustment since it deviates from the standard 3m libor paid quarterly that's quoted on the market. The 3m libor paid in arrears is just an issue of the timing of the date on which refixing occurs. This does not require a convexity adjustment.

>The 3m libor paid in arrears is just an issue of the timing of the date on which refixing occurs. >This does not require a convexity adjustment. Is this really true? I am pretty sure I have seen a few textbooks on the subject discussing convexity adjustments for in-arrears swaps.