Choosing Short-Rate Models with Nonnegative or Negative Rates
Summary
The document surveys discrete-time short-rate models and whether their structure permits negative rates. It lists Cox–Ingersoll–Ross, Black–Derman–Toy, Black–Karasinski, exponential Vasicek, and Hull–White among models designed to keep rates nonnegative, and Ho–Lee and Vasicek among models that allow negative values. Mean reversion is described as movement toward a target rate, though model properties and assumptions differ.
The discussion explains why an unbounded or negative-rate model may still be useful: a derivative’s value can be driven mainly by scenarios with high rates, making its behavior at very low rates less relevant to that payoff. A put on a bond is given as an example, since rising rates depress bond prices and create the payoff of interest. The answer cautions that models have shortcomings; for instance, the cited Black–Derman–Toy mean reversion depends on volatility decay. Model choice therefore depends on the product and the rate behaviors that matter, and the document does not compare calibration performance or provide a general solution for ruin-probability analysis.
Key ideas
- Short-rate models differ in whether their structure allows rates to become negative.
- Mean-reverting models describe rates moving toward a target level, subject to each model’s assumptions.
- A model allowing negative rates can still be useful when the derivative payoff is driven mainly by high-rate scenarios.
- A bond put illustrates how rising rates can dominate the payoff-relevant behavior.
- Model limitations, including assumptions about volatility, matter when choosing a rate process.
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Full text
# Do people use unbounded interest rate models, and what alternatives exist?
# 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:
- Are there any interest rate models in discrete time which assume bounded interest?
- Why do people use models with unbounded interest (which is unrealistic) at all?
## Answer by strimp099 (score 8, accepted)
https://quant.stackexchange.com/a/1952
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.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.