Why Nonstandard Swap Payments Require Convexity Adjustments
Summary
The document explains why a swap whose floating-rate payment timing differs from the standard reset-and-payment convention may need a convexity adjustment. A vanilla swap can be viewed as a sequence of forward rate agreements: forward rates from the zero curve determine the floating payments, which are then discounted on that curve. For an arrears swap, payment timing differs from the FRA convention, so that replication argument no longer applies directly and the measure implicit in the valuation may be mismatched.
The answers offer intuition rather than a full derivation. One connects the adjustment to the nonlinear relationship between bond prices and yields, noting that the effect grows with maturity. Another compares forward and futures rates: futures gains and losses are realized over time, and reinvestment at changing rates can make their rate differ from a forward rate. These analogies help explain the adjustment, but the document gives no formula, calibration method, or quantitative example for a particular swap.
Key ideas
- A vanilla swap can be valued as a series of forward rate agreements using the zero curve.
- When payment timing differs from the FRA convention, that replication argument may fail.
- The nonlinear relationship between bond prices and yields creates convexity effects.
- Futures and forward rates can differ because futures gains or losses are realized over time and may be reinvested.
- The document gives qualitative intuition but no adjustment formula or worked numerical example.
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# Why is convexity adjustment applied to swap price for a nonstandard swap, in simple terms?
# 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.
## Answer by user12348 (score 4)
https://quant.stackexchange.com/a/21643
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.
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/25278
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.