# 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?

• Current rate minus previous rate? – Chris Taylor Aug 9 '17 at 17:04
• how about proportional change? i am trying to estimate tail risk and am hoping to generate time series that follows normal distribution – 291890964 Aug 9 '17 at 17:17
• Proportional changes don't make sense if rates can be zero or negative. – Chris Taylor Aug 9 '17 at 18:06
• Yes, sir. Hence my question. How do I deal with this? – 291890964 Aug 9 '17 at 18:16
• Use current rate minus previous rate. – Chris Taylor Aug 9 '17 at 19:42

## 1 Answer

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.