ATM interest rate swap dv01 vs off-market swap dv01

Question

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

Answer 1

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?

Answer 2

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.