Converting Zero-Rate DV01 into Par Market-Rate Sensitivity
Summary
The document explains the meaning of par sensitivity for interest rate risk under ISDA SIMM. A par sensitivity measures how a portfolio's value changes when the market quotes of the instruments used to build a curve are shifted, rather than when the derived zero rates themselves are shifted. For a swap curve, this means bumping the underlying market rates, not simply changing the swap's fair rate until its net present value is zero.
Because pricing systems commonly value instruments using zero rates, the document describes calculating sensitivity to those rates first, then applying an inverse Jacobian to convert it into sensitivity to the market quotes. A second answer describes sensitivity as repricing after a one basis point risk-factor bump. The discussion is specifically about curve risk and SIMM conventions; it does not provide implementation details for constructing or inverting the Jacobian.
Key ideas
- Par DV01 is sensitivity to market quotes of the instruments used to construct a curve.
- Pricing functions often use zero rates, so their sensitivities may need conversion to market-rate sensitivities.
- An inverse Jacobian maps zero-rate sensitivities to sensitivities with respect to market rates.
- The document frames SIMM sensitivity as a repricing impact from a one basis point risk-factor shift.
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# ISDA SIMM swap sensitivities
# ISDA SIMM swap sensitivities
Most of the commercial SIMM models require sensitivities to be passed in in CRIF format. The documentation mentions that "par sensitivities" need to be used. What exactly is a par sensitivity? When we calculate swap DV01, we bump up/down the market quotes of the underlying swap curve instruments by 1 bp. By par sensitivity, do they mean we need to bump up/down the "fair rate" (which will make NPV of the swap zero) on the swap? Thanks in advance.
## Answer by byouness (score 1, accepted)
https://quant.stackexchange.com/a/46554
Usually, zero curves, that is curves of zero rates are constructed from market instruments having corresponding market rates. For example, a 3M curve will be constructed from 3M instruments.
The SIMM simply states that when computing the DV01 wrt to a given curve, one should shift the market rates (meaninf rates of instruments used to construct the curve) and not the zero rates.
Practically though, one would shift the zero rate as it is what the pricing functions rely on, and then use a Jacobian matrix to convert this sensitivity to the zero rates into a sensitivity to market rates.
In mathematical terms, denoting $z$ the zero rates and $r$ the market rates:
$$ \underbrace{\left[\frac{\partial V}{\partial r(\tau_j)}\right]_j}_{\text{Par DV01}} = \overbrace{\left[\frac{\partial r(\tau_i)}{\partial z(\tau_k)}\right]_{k, i}^{-1}}^{\text{Inverted Jacobian matrix}} \underbrace{\left[\frac{\partial V}{\partial z(\tau_k)}\right]_k}_{\text{Zero DV01}} $$
Usually you have this pricing function $V$ (e.g. to price a swap you have formulas relying on the zero rates, not on the market rates) and not the one relying on the market rates.
## Answer by sw89 (score 0)
https://quant.stackexchange.com/a/46551
Within the SIMM model and particularly for Interest rate and credit, a sensitivity is defined as the following :
S = 𝑉(𝑥 + 1bp) − 𝑉(𝑥)
Where V(x) is the value of the instrument, given the value of the risk factor x (risk factor is particular yield, ex: 3 month LIBOR).
So by sensitivity SIMM is asking you to bump the underlying by 1bp and re-price your portfolio to check the impact.
It's all in here https://www.isda.org/a/zSpEE/ISDA-SIMM-v2.1-PUBLIC.pdfShown 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.