# One factor short rate model

I know one factor model assumes that one stochastic factor can explain the future evolution of all interest rates.

Can someone tell me what is the one factor in economic meaning in the one-factor rate model? Does this stochastic factor has economic meaning?

• $$dr_t=\mu(t,r_t)dt+\sigma(t,r_t)dW_t\,\,?$$ – user16651 Jul 11 '16 at 18:19
• i know this formula. Can you please explain what does the factor stands for? – Quant2015 Jul 11 '16 at 18:37
• $\mu\,?$ , $\sigma\, ?$ or $W_t\, ?$ – user16651 Jul 11 '16 at 18:39
• Wiener driver (Wt) – Quant2015 Jul 11 '16 at 18:48
• Ok Wait......... – user16651 Jul 11 '16 at 18:49

Firstly Vasicek (1977) proposed the stochastic process for the short rate $r_t$ under the physical measure to be governed by the Ornstein–Uhlenbeck process $$dr_t=\kappa(\theta -r_t)dt+\sigma \color{red}{dW_t}\tag 1$$ where $dW_t$ is a White noise or the differential of the Wiener process. In economic time series, the white noise series is often thought of as representing innovations , or shocks . That is, $dW_t$ represents those aspects of the time series of interest which could not have been predicted in advance.
The process $(1)$ is sometimes called the elastic random walk or mean reversion process.The instantaneous drift $\kappa(\theta -r_t)$ represents the effect of pulling the process toward its long-term mean $\theta$ with magnitude proportional to the deviation of the process from the mean. The mean reversion assumption agrees with the economic phenomenon that interest rates appear over time to be pulled back to some long-run average value. To explain the mean reversion phenomenon, we argue that when interest rates increase, the economy slows down and there is less demand for loans; this leads to the tendency for rates to fall.Indeed $$\mathbb{E}\left[ {{r}_{T}}|{{r}_{t}} \right]=\theta +({{r}_{t}}-\theta ){{e}^{-\kappa (T-t)}}$$