Par-Rate and Zero-Rate Delta for Interest Rate Swaps

Summary

The document distinguishes two ways to express interest rate sensitivity for a fixed income instrument such as an interest rate swap. Delta is a linear estimate of how much an instrument's price changes after a small move in the relevant rate; the discussion assumes a parallel shift of the interest rate curve.

Market rates may be represented as par coupon rates or as zero rates obtained through bootstrapping. Since either representation can be used to price the instrument, sensitivity can be calculated in either set of rates: a delta based on zero rates is called zero delta, while one based on par or coupon rates is par delta. The distinction is therefore the rate curve representation used for the sensitivity calculation. The document offers no example, formula, or guidance on choosing between the measures, and its parallel-shift assumption limits the description of curve risk.

Key ideas

• Delta approximates price sensitivity to a small change in interest rates.
• The explanation assumes a parallel shift of the rate curve.
• Par rates and bootstrapped zero rates are alternative representations used to price fixed income instruments.
• Zero delta uses zero rates for the sensitivity calculation, while par delta uses par or coupon rates.

Tags

# What is the difference between par delta and zero delta?


# What is the difference between par delta and zero delta?












I was looking at different methods of calculating delta for Interest Rate Swaps(IRS) and came across the words par delta and zero delta.

I am not sure of the difference between the both and when to use a particular delta.

Need some guidance on this.

## Answer by Dr_Be (score 5, accepted)

https://quant.stackexchange.com/a/22904

Delta is a linear approximation of the change in price due to a small move of the relevant interest rate. Typically a parallel move of the whole interest curve is assumed here. This applies to all kind of fixed income instruments, in particular IRS.

Interest rates can be given as coupon rates (these are the so called par rates, based on prices observable in the market) or zero rates (as a result of a so called bootstrapping process). Both can be used to calculate the price of an instrument so for both types a delta can be calculated. So, if you use zero rates for your linear approximation the result is a zero delta. Same is true for coupon/spot rates.

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.