Interpreting Call Option Duration Through Delta Elasticity

Summary

The document explains an approximate way to relate the duration of a call option on a fixed-income instrument to the duration of the underlying bond. It scales the bond’s duration by the option’s elasticity, calculated from the option delta, underlying price, and option value. Elasticity measures the option’s percentage price response to a percentage move in the underlying, so it captures the leverage embedded in the option.

Under this first-order interpretation, the option’s rate sensitivity is the underlying duration multiplied by that leverage factor. The discussion identifies the expression as an approximation and offers intuition rather than a derivation from a full pricing model. It does not provide assumptions about rates, volatility, or changing delta, nor does it compare the estimate with a numerical example. The relationship is therefore a useful local sensitivity approximation, not a complete account of option risk across large market moves.

Key ideas

• The call’s duration is approximated by scaling the underlying bond’s duration by option elasticity.
• Option elasticity expresses the percentage change in option value for a percentage change in the underlying price.
• Delta, underlying price, and option value determine the elasticity scaling factor.
• The approximation reflects leverage and is a first-order sensitivity relationship.

Tags

# How to derive and interpret the duration of a call option?


# How to derive and interpret the duration of a call option?












I read here that CFA students are taught that

$$ D_{C} = \frac{\Delta_{C} D_{B} B}{C} $$

Where $D$ is the duration, $\Delta_{C}$ is the first derivative of the options price with regards to the price of the underlying, $C$ is the price of a European call option on an underlying fixed income instrument, such as Bunds, and $B$ is the spot price of the underlying.

I have not been able to find a reference to this in Hull (2018) OFOD, or online, other than a reference to the Schweser CFA books "30f - fixed income portfolio management II", which I do not have access to.

## Answer by Alex C (score 5, accepted)

https://quant.stackexchange.com/a/44065

This is an approximation (to first order) based on the idea that the option gives you access to the underlying, but with leverage.

Let the duration of the underlying be $D_B$.

The expression $\lambda=\Delta_c\frac{B}{C}$ is called the elasticity of the option (link), defined as "the percentage change in option value per percentage change in the underlying price".

Consequently the option reacts to changes in interest rates more/less than the underlying does, using $\lambda$ as a scale factor.

So $D_c= \lambda D_B = \Delta_c\frac{B}{C} D_B$

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