Computing Floating-Rate Note DV01 with a Synchronous Curve Shift
Summary
The document explains how to measure a floating-rate note’s sensitivity when both its discount curve and coupon curve move together. It models the curves as a shared market level plus separate baselines, so the price sensitivity to that level is the sum of the sensitivities to each curve. A central finite difference can approximate the result by repricing after shifting both curves up and down by the same amount.
This joint shift has a clear interpretation as exposure to a synchronous market move. The proposed norm of a two-component shift is not needed to define that risk measure; the denominator follows from the change in the single shared level. The discussion gives a mathematical derivation, but no numerical valuation example, and the result depends on the assumption that both curves move by the same amount. Other curve shock patterns would represent different risk measures.
Key ideas
- A shared market-level move shifts the discount and coupon curves together.
- The price sensitivity to that shared move equals the sum of the sensitivities to each curve.
- A central difference can estimate the sensitivity by repricing under matching upward and downward shifts.
- The risk measure depends on the assumed relationship between the two curve movements.
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# Taking a directional derivative to compute the floating rate note DV01
# Taking a directional derivative to compute the floating rate note DV01
I would like to compute the floating rate bond (FRN) DV01 applying the central finite difference.
$DV01 = \frac{NPV(Disc \,+ 0.0001, \,Coupon \,+ 0.0001) - NPV(Disc \,- 0.0001, \,Coupon \,- 0.0001)}{2\times||<0.0001,\,0.0001>||},$
where $Disc$ – discount curve, $Coupon$ – coupon curve, $||v||$ – vector norm.
The numerator is easily resolved and provides some value in the instrument currency. However in the denominator I receive: $2\times||<0.0001,\,0.0001>|| = 2\times\sqrt{0.0001^2 + 0.0001^2} = 2 \times \sqrt{2.0\text{E}-8}.$
I have never seen it in the literature. Is it the correct result or taking $||v||$ is a bad idea, because the discount and coupon curves do not exist in the same "space"?
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/85389
It appears that you are trying to obtain some measure of risk to a generic market movement.
Label your discount curve, $d$, and your coupon curve $c$, and let us suggest that there is some latent market level $m$ such that:
$$ d = \bar{d} + m $$ $$ c = \bar{c} + m $$
The NPV, $P(c, d)$, is dependent upon the coupon curve and the discount curve. Thus we have that:
$$ \frac{\partial P}{\partial m} = \frac{\partial P}{\partial d}\frac{\partial d}{\partial m} + \frac{\partial P}{\partial c}\frac{\partial c}{\partial m} = \frac{\partial P}{\partial d} + \frac{\partial P}{\partial c} $$
The central difference of this is:
$$ \frac{1}{2\delta} \left ( P(d+\delta, c) - P(d-\delta, c) + P(d, c+\delta) - P(d, c-\delta) \right )$$
which reflects the change in NPV when both curves move up synchronously by the same amount.
The central difference of $m$ directly is:
$$ \frac{1}{2 \delta} \left ( P(d+\delta, c+\delta) - P(d-\delta, c-\delta) \right ) $$
becuase of the formula stated above for $d, c$ in terms of $m$.
If you scale this to basis points of risk then you multiply by $\frac{1}{0.0001}$. If your choice of $\delta$ is 0.0001 then you recover your above formula.Shown 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.