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Approximate Bond Price Volatility from Yield Volatility

Article Quant Q&A · Author: Stu
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Summary

The document addresses how changes in a short-term interest rate’s yield volatility relate to price volatility, while questioning whether a proposed bond formula applies to a short-term interest rate option. The response gives the standard first-order duration relationship: percentage price change is approximately negative modified duration multiplied by the yield change.

An example illustrates the approximation for a bond: when yield rises, price falls in proportion to modified duration. This is a local sensitivity relation between yield changes and bond prices, not a complete volatility conversion model for every short-term interest rate instrument. The discussion does not establish how to adapt duration to a particular STIR option, whose contract and pricing conventions may require further modeling. Its evidence consists of the formula and a worked bond example, without broader validation.

Key ideas

  • Modified duration approximates the proportional price sensitivity to a yield change.
  • The relationship carries a negative sign because bond prices generally move opposite to yields.
  • The provided formula is a first-order bond approximation, not a demonstrated STIR option conversion.

Tags

Fixed incomeVolatilityDerivatives pricing
Full text
# What equation will convert implied yield volatility to implied price volatility?


# What equation will convert implied yield volatility to implied price volatility?












I am trying to figure out how to turn implied yield volatility of a short-term interest rate into implied price volatility. Is there an equation to do this?

I have come across the equation for a bond: (price_vol = yield_vol*modified_duration*forward_yield) But, I do not believe this is correct for a STIR option.

## Answer by DanielPNewman (score 1)

https://quant.stackexchange.com/a/46688

percent price change ≈ −modified duration × yield change

Example

- Consider a bond whose modified duration is 11.54 with a yield of 10%.

- If the yield increases instantaneously from 10% to 10.1%, the approximate percentage price change will be:

−11.54 × 0.001 = −0.01154 = −1.154%.

Source

https://www.csie.ntu.edu.tw/~lyuu/finance1/2008/20080227.pdf

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.