Interest Rate Swap Delta ladder, under OIS Discounting

Question

I've been looking on some information when it comes to vanilla Interest Rate Swaps, and building delta ladders under a multicurve environment. IR Swaps - Curve sensitivity at maturity node, this answer has been wonderful in explaining the methods for calculating the PV impact of shifting market instruments. However, I wanted to clarify those calculations when we are using OIS discounting.

We will continue to use the same notation as the linked question. Mainly $Z(t_i)$ refers to the OIS disoucnt factor at time $t_i$ and $S(t,T)$ is the par swap rate for a vanilla IRS with tenor $T$

Under these conditions suppose we want to examine that effect that a 1bp shift of the market swap rate would have on our IRS portfolio. Logically I would assume that $\frac{\partial Z(t_i)}{ \partial S(t,T^{*})} = 0$ because the $Z(t_i)$ refer to OIS discount rates which generally shouldn't be affected by $S(t,T^{*})$ since those rates are not used in bootstrapping OIS discount factors (I am aware that under certain simultaneous methods of yield curve construction there might be a small impact?).

Thus we should have ${\partial V(t)}/{\partial S(t,T^{*})} = \frac{\partial S(t,T)}{\partial S(t,T^*)} \sum_{i=1}^N Z(t_i)$. And furthermore when $T^* = T$, ${\partial V(t)}/{\partial S(t,T^{*})} =1$ thus we should have ${\partial S(t,T)}/{\partial S(t,T^{*})} = \sum_{i=1}^N Z(t_i)$ when $T = T^*$.

Would someone be able to verify these results, or in turn shed some more light on delta ladder construction? Thanks!

Answer 1

This depends very much on the instruments used and the methodology to construct the multi-curve.

For example common practice would be to use IRS and Single Currency Basis swaps to simultaneously solve for the OIS curve. In that scenario changing IRS rates would directly impact the OIS curve, whilst maintaining the same separation via the constant basis swaps.

In my opinion it is a poor metric to have curve construction that keeps OIS curve constant when IRS rates rise and fall and that will lead to many spurious risks that are not best reflective of the way to hedge your portfolio.

What I have done in the past, for theory building, is to use a fundamental set of instruments, i.e. r_1, r_2 are the forward IBOR interest rates, all independent, and then a basis between those IBOR rates and the OIS rates for the same forward period, i.e s_1, s_2.

You can model all calculations using the generally simpler:

$$ \frac{\partial V}{\partial r_i}, \frac{\partial V}{\partial s_i} $$

and then you obtain other quantities via Jacobian transformations:

$$ \frac{\partial V}{\partial S(t, T)} = \frac{\partial V}{\partial x_i} \frac{\partial x_i}{\partial S(t,T)} $$

where $\frac{\partial x_i}{\partial S(t,T)}$ measures how one of my fundamental instruments varies in you own curve model as $S(t,T)$ fluctutaes. And this will be dependent upon the whole construction model, i.e. which other instruments other that $S(t,T)$ make up your curve construction process and how it is interpolated etc.