Modeling Interest Rates That Can Become Negative

Summary

The document compares two ways to model interest rate changes when rates may be positive or negative. A normal, or Bachelier, model treats changes as independent of the current rate level. A shifted lognormal model instead makes changes depend on the rate plus a chosen shift, retaining some relationship between change size and rate level while permitting negative rates down to the negative shift.

The discussion gives model forms and explains their qualitative differences, but provides no data comparison, calibration procedure, or evidence favoring one approach. The shift is described as a convention or a value fitted to data, so its choice can affect model behavior. The appropriate specification depends on the rate series and modeling purpose; the document does not discuss how to select or validate a model in practice.

Key ideas

• The normal model makes rate changes independent of the current rate level.
• A shifted lognormal model scales changes with the rate plus a chosen shift.
• The shift determines how far below zero the shifted model can represent rates.
• The document offers no empirical comparison or model selection guidance.

Tags

# Modeling Interest Rate Time Series


# Modeling Interest Rate Time Series












What is the right way to express the change in interest rate time series, if this time series contains both positive and negative rates?

## Answer by Adam N. (score 1)

https://quant.stackexchange.com/a/35541

The usual approaches used to deal with negative interest rates are:

a) the normal (Bachelier) model or Brownian motion, where $dr_t = \sigma dW_t$; this makes changes independent from the level of the interest rate,

b) shifted lognormal (displaced diffusion) model, where, instead of the ordinary Geometric Brownian Motion $dr_t = r_t \sigma dW_t$, we have $dr_t = (r_t + h) \sigma dW_t$ with for example $h=2\%$ or any similar value accepted by convention or made to fit the data; this keeps changes somewhat proportional to the rate level, while allowing negative values up to $-h$.

You can also look at this question, which has useful references.

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