# Convexity adjustment for a forward swap rate

I recently heard that for a forward swap rate (for example, the fixed rate of a swap that will start in one year and end in five years), I need to do a convexity adjustment in order to get the right number. Is it true, and if so why?

• The question in this form is incomplete. The swap rate alone does not need any convexity adjustment. You have to specify how this rate is paid. – Christian Fries Apr 26 '13 at 18:16

First of all it is not clear what exactly you mean by right number, you definitely do not adjust forward swap rate.

You probably mean adjusting euro dollar futures contract rates so that you can later use these values to fit the swap/forward libor curve.

Reason for adjustment is simple. If you are short ED futures and rates go higher futures price drops and you make money. Clearing house of exchange reimburses you excess margin and you can reinvest them at higher rate. If rates go lower you have to put extra cash into your margin account, you can borrow this money at lower rate.

Assuming some distribution and forward rate model(and here things vary from simple vasicek model to extremely complicated example Peterbarg) one can compute how much this advantage is in dollars and convert it to basis points.

Pure intuition tells that longer is the expiry higher is the adjustment also higher is the vol higher is the adjustment.

So the end result is that being short gives you advantage and market participants are aware of it and penalize short side by that amount.

• The convexity adjustment needed for futures comes from the margining applied to the (undiscounted) future price. In contrast, swaps are collateralized by discounted value, such that a future-like convexity adjustment does not apply. However, if a forward swap rate is paid in an unnatural way (like in a CMS), a convexity adjustment applies. – Christian Fries Apr 26 '13 at 18:45

Given an index $t \mapsto S(t)$ (this may be a forward swap rate) and some value process $t \mapsto A(t)$ (this may be a swap annuity) we assume that $S/A$ is a traded product (which is true if $S$ is the forward swap rate and A is the corresponding (!) swap annuity. Then the future payoff $S(T) \cdot A(T)$ can be values as $S(t) \cdot A(t)$ (since $S$ is a martingale under the measure $Q^A$).

Now, if we consider the payoff $S(T) \cdot P(T)$ (say for example if $P$ is the zero coupon bond with maturity $T$) then the value can be expressed as $S'(t) \cdot P(t)$ where $S'(t)$ is the so called convexity adjusted rate, that is $S'(t) = E(S(T) \cdot \frac{P(T)/P(t)}{A(T)/A(t)})$ (with expectation under $Q^A$). The convexity adjustment is a correlation term coming from the correlation of the index to the change of the payment.

That said: If you need a convexity adjustment depends on how the index is paid.

Most likely the question is about CMS rate convexity adjustment. i.e. today value of a swap rate that fixes at some future time T.

Mathematically, the adjustment arises from different measures (annuity versus forward measure).

This is a good reference http://www.math.nyu.edu/~alberts/spring07/Lecture4.pdf

As a rule of thumb, the size of the adjustment depends on

1. CMS maturity
2. Level of vol
3. skew

For the latter the classical reference is http://www.gorillasci.com/documents/convexity.pdf

• gorillasci.com/documents/convexity.pdf is no longer available. Your other reference is quite good and bold in presentation. – user12348 Oct 28 '15 at 1:38
• it's "Convexity conundrums: Pricing CMS swaps, caps, and floors" paper by P.Hagan – alex Dec 13 '18 at 3:54

Yes, an adjustment has to be made and the reason is that a forward curve now will evolve and not be the same as the future spot curve. For example, a one year forward today is not equal to your spot rate a year hence. So spot curve discount factors have to be adjusted or directly replaced through the forward DFs.