Choosing Interest-Rate Distributions and Modeling Rate Dynamics
Summary
The document compares distributional choices for modeling interest rates, beginning with a proposed sequence of nonnegative independent observations. Responses caution that this assumption misses persistence and changing volatility: rates tend to cluster and, in many developed markets, exhibit mean reversion. For a simplified case study, a lognormal model is suggested, while a Cox–Ingersoll–Ross process offers a mean-reverting alternative with noncentral chi-squared innovations.
Other replies point out that normal models can fit low or negative rates and are reflected in some swaption volatility conventions. Gaussian affine term-structure models, empirical distributions fitted to relevant data, and a triangular distribution are also mentioned. These are suggestions rather than a comparison backed by fitting results. The appropriate choice depends on the market, sample, and purpose; in particular, nonnegative distributions cannot represent negative rates, and a distributional assumption alone does not capture realistic serial dependence unless embedded in a dynamic model.
Key ideas
- Interest rates commonly show persistence and mean reversion, so an independent identically distributed model may be unrealistic.
- A lognormal distribution is a simple choice for positive rates but permits rates to grow without bound.
- The Cox–Ingersoll–Ross process models mean reversion and uses noncentral chi-squared innovations.
- Normal models can accommodate low or negative rates and appear in some rate-option quoting conventions.
- Empirical fitting and market context can guide distribution choice.
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Full text
# What distribution to assume for interest rates?
# What distribution to assume for interest rates?
I am writing a paper with a case study in financial maths. I need to model an interest rate $(I_n)_{n\geq 0}$ as a sequence of non-negative i.i.d. random variables. Which distribution would you advise me to use? Currently I am considering the exponential distribution, but I am not sure that it is the right choice, though it is quite easy to work with.
## Answer by Maxareo (score 6, accepted)
https://quant.stackexchange.com/a/2338
Exponential distribution, although it's a good distribution for modeling non-negative numbers, doesn't make sense here since it's mode is 0.
From a pure statistical point of view, without any knowledge of interest rate, I'd recommend log-normal as in modeling stock prices and inverse-gamma or gamma distribution which are used to model variance or other scale parameters which is a non-negative distribution with mode greater than zero.
## Answer by Tal Fishman (score 14)
https://quant.stackexchange.com/a/2429
Interest rates in general are far from independent and identically distributed. A high interest rate observation is quite likely to be followed by another high observation, and the volatility is likely to be higher as well. Interest rates are also mean reverting, as in most real-world situations (at least for developed markets) interest rates rarely rise too high or dip too low.
Since you are looking for the simplest possible solution for a case study, I would recommend you start with a lognormal distribution, which implicitly assumes interest rates follow a geometric brownian motion. The problem with this distribution is that it assumes the interest rate can get arbitrarily high. The next simplest solution would be a Cox-Ingersoll-Ross process, which has a noncentral chi-squared distribution of innovations. The following matlab function includes a simple simulation of a CIR process. The underlying distribution the function uses is noncentral chi-square, and the algorithm itself is quite clear even if you don't use or know matlab.
## Answer by user7056 (score 7)
https://quant.stackexchange.com/a/4033
You could try using the Gaussian Affine Term Structure Models (GATSM), with the right boundary conditions to stop rates being negative (in the style of their Black implementation). See, for example, Monika Piazzesi, the "Affine Term Structure Models" if you want to enter/modify the basis or the work of Krippner, for example "Measuring the stance of monetary policy in zero lower bound environments".
## Answer by Strange (score 3)
https://quant.stackexchange.com/a/4100
General rate trading wisdom shows that if anything, normal distribution fits developed markets better. For example, most swaption traders talk about implied volatilities in basis points (per day or annualized).
## Answer by Randor (score 3)
https://quant.stackexchange.com/a/19498
Normal distribution makes most sense these days for ratesthat are very low, or even negative, like euribor, chf libor
Normal distribution is what is assumed by option brokers impliedvolatility quotes for these currencies
## Answer by Babar (score 2)
https://quant.stackexchange.com/a/2354
If you manage to get some data fitting your subject, one solution could be to try an empirical distribution.
## Answer by TVC (score 1)
https://quant.stackexchange.com/a/74822
I use the triangular distribution as these rates, treasury or corporate bonds, look like a triangular to me.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.