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Modified Duration of Treasury Futures and Their Delivery Options

Article Quant Q&A · Author: A1122
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Summary

The document explains how to estimate a Treasury futures contract’s modified duration when its cheapest-to-deliver (CTD) bond is known. Under the simplifying assumption that futures track the CTD’s forward price perfectly, the conversion factor cancels from the duration calculation, so futures modified duration matches the CTD’s forward modified duration. This clarifies why a calculation based on futures DV01 and price can produce a duration similar to the cash CTD’s.

That result depends on delivery conditions. When one bond is overwhelmingly likely to be delivered, the approximation may be reasonable. If several bonds compete to be CTD, delivery options can change the futures’ rate sensitivity, making the current CTD’s duration an unreliable proxy. The document recommends modeling delivery options and measuring option-adjusted DV01 or duration in those cases; it does not provide a worked model or empirical comparison.

Key ideas

  • With perfect tracking, futures modified duration equals the CTD’s forward modified duration.
  • The conversion factor cancels when futures price is expressed as the CTD forward price divided by that factor.
  • A single, highly likely CTD supports the simplifying duration approximation.
  • Competing delivery candidates can make futures duration differ substantially from the current CTD’s duration.
  • Delivery-option modeling can produce more appropriate option-adjusted rate-risk measures.

Tags

FuturesFixed incomeDerivatives pricingRisk management
Full text
# Modified duration of treasury futures tracking CTD?


# Modified duration of treasury futures tracking CTD?












If I know TYU7 contract's CTD is T 2.500 05/15/2024 with modified duration of 6.37. I know futures DV01 is calculated by taking the CTD's DV01 divided by conversion factor as shown here. What is the modified duration of TYU7?

I tried backing out the Mod Dur based on the formula for calculating Mod Dur using DV01 for cash instruments, which is $$Mod Dur = \frac{DV01}{0.01*0.01*Price}$$

Given the future's DV01 and its price, this gave me a Mod Dur for futures which is the same as Mod Dur for cash CTD. Kinda weird, isn't it?

## Answer by Helin (score 8, accepted)

https://quant.stackexchange.com/a/34554

Let's make a simplifying assumption that futures perfectly track their CTDs, then

$$ D_\text{mod, fut} = \frac{1}{f}\frac{df}{dy} = \frac{1}{F_\text{CTD} / \lambda_\text{CTD}} \cdot \frac{dF_\text{CTD} / dy}{\lambda_\text{CTD}} = \frac{1}{F_\text{CTD}}\frac{dF_\text{CTD}}{dy}, $$ where $f$ is the futures price, $F_\text{CTD}$ is the CTD's forward price, and $\lambda_\text{CTD}$ is the CTD's conversion factor. So you're right; under the assumption that futures track its current CTD, then its modified duration is identical to the CTD's (forward) modified duration.

However, I cannot emphasize how dangerous the assumption can be. In the current environment where CTDs have 100% delivery probability into the futures, you're fine. But in other environments, multiple underlyings can be contenders for CTD status. In these cases, futures duration can be very different from the current CTD. The proper thing to do is to build a futures model that account for delivery options and calculate risk metrics such as option-adjusted DV01/duration.

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.