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.
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2 Answers
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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.
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$\begingroup$ Thank you. I knew there was more to it than just computing DV01 as usual. Do you know of any references/examples/book that shows how the Jacobian is constructed? $\endgroup$ Jul 11, 2019 at 17:57
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$\begingroup$ The market rate of maturity $\tau_i$ will depend on the zero rates up to maturity $\tau_i$ leading to a triangular Jacobian matrix. Just like any derivative, the terms of the Jacobian matrix could be computed by (1) finite differences, by using the formulas giving the market rates as functions of the zero rates, (2) analytically if you manage to differentiate these fromulas, or using algorithmic differentiation. I don't have any references really, but after googling I found this paper which looks to provide some insights: uglyduckling.nl/library_files/PRE-PRINT-UD-2014-02.pdf $\endgroup$– byounessJul 11, 2019 at 20:02
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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.pdf
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$\begingroup$ Okay, so what you are suggesting is the usual approach for calculating IR sensitivity. If you take a look at slide#3 at link, it says "2Y Swap Rate and not 2Y Zero rate". I am assuming they are talking about the instruments used to build the swap curve? $\endgroup$ Jul 11, 2019 at 17:05