# What is the right group of durations?

It seems that the group of durations commonly used in quantitative analyse is $\mathbf{R}$ but it seems to me that $\mathbf{R_+^*}$ could also be an interesting choice.

While I am not aware of explicit references to $\mathbf{R}$ as the group of durations, there is a few implicit hints in quantitative literature that this choice has been made.

1. The elementar theory of interest rates is constructed around the exponential, that morphism from to $\mathbf{R}$ to $\mathbf{R_+^*}$, thus making the implicit statement that the group of durations is $\mathbf{R}$.

2. The usual modelisation of the volatility in LIBOR market models can be written in terms of the first Hermite functions (forgetting about the constant term) which are the eigenfunctions of the Fourier transform on $\mathbf{R}$.

Choosing $\mathbf{R}$ as the group of durations has shown to be pertinent in deterministic settings, like classical mechanics. We are however sudying information and stochastic processes, which introduce a great assymetry between the past and the present on one side and the future on another side: as a practical consequence, the group structure on $\mathbf{R}$ is more an accident than anything else, because there is no action of the group on the information we are considering.

It seems however that choosing $\mathbf{R_+^*}$ as a group of durations might be interesting:

1. The group $\mathbf{R_+^*}$ operates on lognormal processes by rescaling the volatilities (with a square root). If $R$ is the vector space of allowed coefficient functions for Itô processes, we have a seemingly interesting operation of $\mathbf{R_+^*}$ on $(Rdt + RdW)/Rdt$.

2. The group $\mathbf{R_+^*}$ also can operate on forward rates, i.e. on maturities.

Did anybody explore this road? Are there any evidence that this could be a useful point of view?

-
Can you define "group of durations"? you don't mean interest rate sinsitivities right? what is the group operation? And in your point 1, you have an mapping of $\mathbb{R}_+^*$ on what space? –  vanguard2k Dec 6 '13 at 9:48
I mean duration as a difference between two time points, i.e. a translation transforming a time point into another. I do not understand your question about point 1., would you mind rephrasing it? –  user40989 Dec 6 '13 at 10:24
Do you mean the set of Ito Processes with constant coefficients? They will be proportional to $e^{R(t+W_t)-R^2t/2}$ of course and I think the "link" gets lost here because of the $R^2$-term. One more point: If you take $R_+^*$, it is only a group with the operation $*$, not with $+$. Semantically, when refering to duration, we measure the difference of time points. Of course, when using $+$, the group property and therefore the exponential as a homomorphism is lost. Thats why I asked about "group of durations"... –  vanguard2k Dec 6 '13 at 11:29