# Do people use unbounded interest rate models, and what alternatives exist?

A simple interest rate model in discrete time is the autoregressive model, $$I_{n+1} = \alpha I_n+w_n$$ where $\alpha\in [0,1)$ and $w_n\geq 0$ are i.i.d. random variables. When working with ruin probabilities in a model which incorporates this interest rate model, I've faced that $I$ can reach any positive value with a positive probability.

Hence I'd like to know:

1. Are there any interest rate models in discrete time which assume bounded interest?

2. Why do people use models with unbounded interest (which is unrealistic) at all?

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– Tal Fishman Sep 15 '11 at 20:47
@Tal Fishman: Nice observation, though most of the models seem to be developed for a kind of "equilibrium" state of economy (like in a mode of growth), so the example with extremely high interest rates won't be enlightened in that models anyway. Btw, thanks for bounty, it worked as appeared. – Ilya Sep 19 '11 at 8:34
Hello @Tal Fishman and Gortaur, has such an auto regressive model forecasted the presently observed negative interest rate bonds, without taking into consideration any influence of the credit risk on this market risk factor (governmental downgradings), taking into account that France and USA have lost their triple-A and Germany is about to loose it? – user7056 Aug 27 '12 at 16:17

## 1 Answer

There are certainly (short-rate) models which assume bounded interest rates. I suppose I should clarify - the design of the model prohibits negative interest rates. Further, some models asymptotically reach some target, or mean rate which is considered mean reversion, the most famous perhaps the Vasicek.

Short rate models where rates cannot go negative:
Cox-Ingersoll-Ross
Black-Derman-Toy
Black-Karasinsky
Exponential Vasicek
Hull-White

Short rate models where rates can go negative:
Ho-Lee
Vasicek

These are all stochastic models that can be solved in discrete time.

Of course, each of these models have there own shortcomings. For example, the mean reversion in the Black-Derman-Toy model is dependent on volatility decay which only happens in practice if the modeled volatility is fit to traded securities whose volatility diminishes over time.

People may use an interest rate model which has the possibility of negative interest rates if they are valuing derivatives that might have a payoff of 0 when rates get low, or below some low but positive value. In other words, the interesting stuff happens at some high positive interest rate.

For a simple example, think of a put option on a bond. This contract only gets interesting when the price of the bond goes down (with the corresponding rates going up) so it really doesn't matter if the model has negative rates in it because we're only interested in the payoffs when rates are high.

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Cool, I will take a look of them – Ilya Sep 19 '11 at 8:35

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It seems that all these models still allow the interest rate to be unbounded. – Ilya Sep 19 '11 at 13:46
@Gortaur I think that is because all the modelers behind these seminal models made the determination that a zero lower bound and no upper bound are realistic assumptions. The put option on a bond is a good example of where all the action is specifically in the very high interest rate scenarios, and it would make little sense to artificially cap interest rates within a model. Since some models even allow rates to go negative, perhaps that makes a zero lower bound seem less extreme to you? – Tal Fishman Sep 19 '11 at 15:05
@TalFishman: I would say that interest rate is something which allows you to receive small but positive profit - otherwise you just do not invest there, but maybe I am missing smth. – Ilya Sep 19 '11 at 15:27
@Gortaur there are extreme situations where even nominal rates can be negative, so positive yet very low rates seem positively plausible (sorry for the pun). – Tal Fishman Sep 19 '11 at 15:31
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