> given that duration expressed as decimal, we can > remove 100, so duration * delta * underlying > price/price of an option = option duration I checked a book called “Swaps and Options” on Google books and this is the equation they have. So it seems that the CFAI forgot the Delta.
yes, they forgot the delta but corrected it http://www.cfainstitute.org/cfaprog/resources/pdf/Level_III_Errata.pdf
Hmm… When the interest rate changes, the underlying changes price. So the underlying has a duration. For small changes in the underlying price, owning an option is similar to owning a fractional portion of the underlying. That fraction is equal to the option’s delta. So if an option is ATM, it’s like owning 1/2 of the underlying. So, the option duration should be: (Delta) * (Duration of the Underlying) The only issue on my mind is that Rho tells you how an option price changes with changes in the RfR. So maybe it’s: (Rho) * (Delta) * (Duration of Underlying). Can anyone clarify here? (PS: Thanks Joey for the comments on solving differential equations; I decided I sucked at math because I could almost never find the trick for solving pdfs. I also sucked at figuring out how to do some of the weird integrals that physicists like to use in field theory. But I could look deep into a client’s eyes and figure out their risk tolerance. 
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Let’s say that delta of an interest rate option includes rho. This whole thing is about approximation and there is no doubt that rho and delta are a little ambiguous when you are talking about interest rate options (One has to do with the interest rate until option expiry and the other has to do with the underlier).