# ATM interest rate swap dv01 vs off-market swap dv01

How significant is impact on dv01 of an at-the-money swap if rates move significantly?

For example: lets say a 5Y USD at the money swap has 4.6 duration. now, if the rate curve move by 150-200bps, what will be the new dv01? Is there a heuristic rule to estimate this?

thanks

To a very (!) first approximation, the delta of a swap equals the negative of its time to maturity ($$\Delta \approx -T$$), the gamma equals that squared ($$\Gamma \approx T^2$$): Starting from a generic swap valuation formula \begin{align} PV&=s\sum_i\delta_{t_i}^{fix}D(t_i)-\sum_i\delta_{t_i}^{float}F(t_{i-1}\to t_i)D(t_i) \end{align} with swap rate $$s$$, discount factor $$D$$, forward rate $$F$$, and year fraction factors $$\delta$$. Let's simplify and assume a single curve world ($$F$$ derived from $$D$$), annual payment frequencies ($$\delta_i=1$$) and payment dates $$t_1=1,t_2=2,\ldots,t_n=T$$

\begin{align} PV&=s\sum_ie^{-r(t_i)t_i}-(1-e^{-r(t_n)t_n})\\ \Rightarrow \Delta \equiv \frac{\partial PV}{\partial r}&=-s\sum_i t_ie^{-r(t_i)t_i}-t_ne^{-r(t_n)t_n}\\ &\approx-t_n\\ &=-T \end{align}

Likewise, the second order derivative, $$\Gamma$$, is approximated as

\begin{align} \Gamma \equiv \frac{\partial^2 PV}{\partial r^2}=\frac{\partial \Delta}{\partial r}&=s\sum_i t_i^2e^{-r(t_i)t_i}+t_n^2e^{-r(t_n)t_n}\\ &\approx t_n^2\\ &=T^2 \end{align}

Then,

$$d\Delta=\Gamma dr\approx T^2dr$$

Do note, however, that this approximation only holds in a low-interest-rate environment. As soon as $$r,c\gg0$$, the approximation deteriorates.

HTH?

• How to think about units? so for example, a 5Y swap is ~4.6 yr duration and lets ball park it as 5. then change in duration aka convexity is T^2 = 25 in what units? are you implying duration will change by 0.25 years for 100bps move? Jun 23 at 18:09
• yes, exactly that. Approximately, delta will change by Gamma (T^2) times the rate move. If we do all the calculations in natural units (as in my post), that's the result. Jun 23 at 21:32

You need Gamma to answer this question really. Gamma tells you how much your delta moves for a change in rates. Taking a 5y \$receiver swap with a DV01 of \$4333.60 on 10MM notional we get a Gamma per 1bp of \$2.41. If we now shift all curves (3m\$L and SOFR) by -150bp this swap becomes more ITM and hence the DV01 should increase. Approximately the new DV01 is now 4333.60 + 150 * 2.41 which gives us \$4695.1. This is not too far off from the actual new DV01 of \$4715.72 when looking at the pricer. I'm not aware of any heuristics without Gamma.

• The only heuristic I can think of is that if you're borrowed (paying fixed) you are short gamma (so the dv01 will get less if rates move up and increase if they move down) and if you're lent (rec fixed) then you are long gamma (the dv01 will decrease when rates move up and increase when they move down). Jun 22 at 11:00