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.
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$\begingroup$ You could try a discretisation of CIR process, which should give you a non central chi-square distribution if I remember well. $\endgroup$– TheBridgeNov 8, 2011 at 13:04
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$\begingroup$ @TheBridge: thanks, but I wonder if there any benchmark distributions for iid interest rates $\endgroup$– SBFNov 8, 2011 at 14:17
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$\begingroup$ What is the best distribution to describe a financial time series is up for serious debate.....Every one will say to use something different $\endgroup$– pyCthonAug 30, 2012 at 19:40
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7 Answers
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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.
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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.
answered Nov 24, 2011 at 18:52
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$\begingroup$ I used the matlab function (translated to R), and it gives quite reasonable numbers. $\endgroup$ Aug 29, 2012 at 6:55
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1$\begingroup$ "Interest rates in general are far from independent and identically distributed" -- a statistician's least favorite phrase, but so very true in this case. $\endgroup$– jlowinAug 29, 2012 at 12:33
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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".
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1$\begingroup$ Hello @Bob Jansen : You could try Monika Piazzesi, the "Affine Term Structure Models" chapter, if you want to enter/modify the basis. Or you can see the work of Krippner, for example "Measuring the stance of monetary policy in zero lower bound environments", if you want to cut to some final ideas. $\endgroup$– user7056Aug 30, 2012 at 11:56
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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).
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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
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If you manage to get some data fitting your subject, one solution could be to try an empirical distribution.
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I use the triangular distribution as these rates, treasury or corporate bonds, look like a triangular to me.
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$\begingroup$ It seems a nice idea, simple to implement, but may not be suitable for option valuation since the tails of a triangular distribution cut off so sharply. We are assuming there is literally zero probability of an interest rate smalller than $a$ or bigger than $c$, but strange things can happen in financial markets... $\endgroup$– nbbo2Mar 4, 2023 at 10:41