Estimating Fixed-Income Convexity from Symmetric Yield Shifts

Summary

The document gives a finite-difference estimate of a fixed-income instrument’s convexity using its current price and projected prices after equal upward and downward parallel yield shifts. The central second difference, formed by adding the two shifted prices and subtracting twice the current price, is divided by the squared yield change and normalized by current price. This approximates the curvature of price with respect to yield.

The method requires projected prices under clearly stated assumptions. The answer specifically notes that spread behavior must be controlled, commonly by holding spreads constant during the parallel shift. It also says a related approach can use duration estimates after small yield shifts. The question asks whether price, DV01, and duration alone determine convexity, but the response does not derive a direct formula from only those inputs; it relies on repricing or shifted-duration information.

Key ideas

• Convexity measures the curvature of instrument price as yield changes.
• A symmetric finite difference estimates curvature from prices under equal positive and negative yield shifts.
• The estimate depends on assumptions used to produce shifted prices, including spread behavior.
• Shifted duration estimates can support a related approximation when direct repricing is unavailable.

Tags

# If I know the Price, DV01, and Duration of a Fixed Income instrument, is their approximation for the Convexity?


# If I know the Price, DV01, and Duration of a Fixed Income instrument, is their approximation for the Convexity?












As the title says, I am looking to see if there is a good approximation for the convexity of a Fixed Income instrument. Say I know all the parameters of the instrument, can the Convexity be written as a function of these?

## Answer by Sharad (score 4, accepted)

https://quant.stackexchange.com/a/75114

Let $P$ represent the current price and let $P(-\Delta y)$ and $P(+\Delta y)$ represent the projected prices if the yield curve is shifted in parallel by the amounts $-\Delta y$ and $+\Delta y$ respectively (we need to exercise care with respect to the assumptions under which these projected prices are obtained, for example one typically assumes that spreads remain constant). Then:

\begin{align*} \mbox{Convexity} &= \frac{1}{P} \frac{d^2P}{dP^2} \\ &\approx \frac{1}{P} \frac{\left(\frac{P(-\Delta y) - P}{\Delta y} - \frac{P - P(+\Delta y)}{\Delta y}\right)}{\Delta y} \\ &= \frac{1}{P} \frac{P(+\Delta y)+P(-\Delta y) - 2P}{(\Delta y)^2} \end{align*}

You can recycle a version of the argument if you have information on the Durations $D(-\Delta y)$ and $D(+\Delta y)$ for small parallel shifts of the yield curve.

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.