# Why is convexity adjustment applied to swap price for a nonstandard swap, in simple terms?

Martinelli et al. show that when the 3-month Libor is replaced by the 3-month Libor forward rates (which are obtained from the spot zero-coupon yield) then the swap price depends only on zero-coupon prices. I argue that this result still true when the difference between the measurement date and the payment date is not equal to the maturity of the 3-month Libor. The Libor is replaced by the forward rate which depends on $B(t,T_{measurement})$ and $B(t,T_{_{measurement}+maturity})$. However the swap can not be regarded as the difference between the price of a coupon-bearing bond maturing at the maturity date of the swap and the price of a zero-coupon maturing at the next floating cash-flow payment. Thoughts, am I on the right track? Furthermore, Martinelli et al. claim that it is necessary to apply a convexity adjustment to the forward rate. I hope some could explain in simple arguments.

• Hi Khaled Bennour, welcome to Quant.SE! Can you provide a reference to the Martinelli et al. article? It will be hard to find for others like this. – Bob Jansen Oct 24 '15 at 7:57
• Hi. It's about the book of Martinelli et al. "Fixed-income securities", Wiley – Khaled Bennour Oct 24 '15 at 8:35
• This cannot be explained in simple argument as it involves measure changes and approximations. Check Brigo, amazon.ca/Interest-Rate-Models-Inflation-4-Aug-2006/dp/…, for discussions – Gordon Nov 9 '15 at 21:04
• I propose to close this question, as two answers are provided, but not a single comment from the OP. – Gordon Jun 1 '17 at 16:42

You understand that forward curve can replicate the payment on FL at the next reset date. Therefore, the vanilla swap can be valued as a series of FRA - meaning, you assume that the forward rate will be realized and the resulting cash flow can be discounted to the present using the zero swap curve.

When you have a non-standard situation, like for an arrears swap the payment is at the time of the reset, then this scheme fails, rather than the next reset date. It fails because the payment is not following the FRA convention. This is in effect, as if a wrong measure has been applied.

As you walk the zero curve on time, the forward rate changes. The relationship between the bond price is yield is a convex function. As the yield changes the price changes in a non-linear manner. This causes convexity effect. Longer the maturity the greater is the convexity effect, hence the convexity adjustment.

For arrears swap, if the frequency of the payment is annual, and it is paid a year earlier, the convexity adjustment is for every year of the swap until maturity. It is difficult for many to get an intuition on because it is usually wrapped up in elaborate mathematical expressions.

A convexity adjustment is often applied to fix the difference between the view of the instrument as it is and if it were based on forward rates.

A simple example is the difference between a futures rate and a forward rate.

The difference is that the forward pays after the period whereas the futures price has a pnl all the way from the purchase up to expiration.

If rates rise the gains on a short position in the futures can be reinvested at a higher rate. Thus the price must be different to a forward where this is not possible. This is the difference between future rate and forward rate.

I guess some similar reasoning of reinvesting applies to your setting too.