Finite-Difference Interest Rate Sensitivities for Curve Instruments

Summary

The document considers two ways to estimate a product’s interest-rate sensitivity to a yield curve built from quoted instruments. One method shifts every instrument quote by the same amount, rebuilds the curve, and revalues the product. The other shifts each instrument separately, measures each impact, and sums those impacts.

For exact derivatives, the total change from moving all quotes together equals the sum of the individual partial derivatives, by the chain rule. In practice, the sensitivities are approximated with finite differences, so the two calculations will rarely match exactly. They should be close when the approximation is suitable; if the discrepancy is material, the answer suggests trying a smaller quote shift. This guidance is qualitative and does not specify an optimal bump size or quantify the finite-difference error.

Key ideas

• The chain rule equates a joint infinitesimal move with the sum of individual partial derivatives.
• Finite-difference estimates of sensitivities need not add up exactly.
• A joint quote shift and summed single-instrument shifts should be close when the finite-difference approximation is adequate.
• Trying a smaller quote shift can help assess whether bump size contributes to a discrepancy.

Tags

# IR risk sensitivity to curve instruments


# IR risk sensitivity to curve instruments












I need to understand if the 2 approaches are equivalent: assume I am constructing a yield curve with N instruments. I would like to compute IR delta for a product using this curve. One approach is to move every instrument's quote by 1bp, reconstruct the curve and revalue. Another one is to do one at a time, compute the impact of that particular instrument and sum the results across. Would those be equivalent? I don't get the same answers and I would like to confirm if this is due to FDM error or there is a math reason for the difference.

## Answer by Kurt G. (score 1)

https://quant.stackexchange.com/a/69572

It sounds like you expect the sum of the single impacts to be equal to the impact you get when you move every instrument by 1bp. In theory this should be the case because (assuming two curve instruments for simplicity which happen to have same levels $x$) $$ \frac{d}{dx}f(x,x)=\frac{\partial}{\partial x_1}f(x_1,x_2)\Big|_{x_1=x_2=x}+\frac{\partial}{\partial x_2}f(x_1,x_2)\Big|_{x_1=x_2=x}\,. $$ In practice however you are approxmating these derivatives by finite differences so that there will rarely be an equality. What you can expect is that the results are close. If they aren't you can try a move smaller than 1bp.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.