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

Question

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 1

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