# What is drift in interest rate term structure model

I was studying about the interest rate term structures and i came across term structure model with (and without) drift.

I am really unsure about what this drift is in this equation for term structure model. $$dr=\lambda dt + \sigma dw$$

From the equation above $\lambda$ is the drift factor and $\lambda dt$ is the drift. I have a very confusing explanation of drift which is along the lines of interest rates are moved in the future by some factor.

Can someone give me an explanation of drift. An example associated with it would be ideal. Thanks!

• Let's face it, empirically single factor models $dr=\sigma dw$ do a poor job of modeling the actual term structure. I believe adding $\lambda$ is a "hack" intended to improve the fit at the cost of model realism (after all interest rates don't always go up). An alternative approach is to go to 2 factor and other models like CIR and HJM. – noob2 Feb 10 '16 at 17:35

## 1 Answer

Many term structure models-both single-factor and multifactor imply dynamics for the short-term riskless rate $r$ that can be nested within the following stochastic differential equation:

$dr = (\alpha + \beta r)dt + \sigma r^\gamma dZ.$

These dynamics imply that the conditional mean and variance of changes in the short-term rate depend on the level of $r$.

On your case we have $\alpha = \lambda$ and $\beta=\gamma=0$, and the model simplifies to the one on Merton (1973). So $\lambda$ is just capturing the growth over time of the interest rate. If there was no uncertainty, it would mean that interest rates would grow forever. Usually we don't see this in the data that's why most models haave a $\beta < 0$ which implies that interest rates are mean reverting.