Why Positive Durations Could Model Time Scaling in Stochastic Finance
Summary
The document questions the usual treatment of duration as an element of the additive real numbers and proposes the positive real numbers under multiplication as an alternative. It observes that interest-rate theory uses exponentials mapping real values to positive values, and that some LIBOR volatility models use functions associated with the Fourier transform on the real line. These are presented as indirect clues about the conventional choice, not as a formal argument that one duration structure is superior.
The proposed alternative is motivated by the asymmetry between past information and future outcomes in stochastic processes. Positive rescaling can act on lognormal processes by changing volatility according to the square root of the scale, and can also act on forward rates through their maturities. The document offers these as potentially useful mathematical structures, but gives no model construction, empirical results, or cited research establishing their value. It is an open conceptual question about the representation of time in quantitative finance.
Key ideas
- Duration is usually represented with the additive real numbers, although the document questions whether that convention is necessary.
- Interest-rate exponentials and some volatility models provide indirect connections to the real-number representation.
- Positive multiplicative scaling can change lognormal-process volatility by a square-root factor.
- The proposed scaling group may also act on forward rates through maturity.
- The document raises a research question and supplies no empirical validation or specific application.
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Full text
# What is the right group of durations?
# 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.
- 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 $\mathbf{R}$.
- 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:
- 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 $(Rdt + RdW)/Rdt$.
- 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?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.