# Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*):

$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$

(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)

and the SDE of Ornstein-Uhlenbeck process:

$$d X(t) = \sigma d B(t) + \theta (\mu - X(t)) d t$$

In which case the one or the other is better suited for modelling financial data? I read that currrency price data can be well modelled by O-U process. Is there a heuristic/empirical argument for that ?

• Geometric Brownian motion is generally used to model stock prices, while the OU process is used for interest rate, or anything that has the mean-reverting nature. – Gordon Jan 22 '16 at 0:58
• @Gordon, apart from Brownian motion, Geometric brownian motion, O-U process, what are the next process I should learn about? (the most famous/used ones) ? – Basj Jan 22 '16 at 15:45
• These are the major ones. The other ones will be Poisson and Cox processes that are used in credit risk modelling. Levy process may also be helpful. – Gordon Jan 22 '16 at 16:06

## 2 Answers

Given efficient markets, asset prices should be unpredictable in the sense that any upcoming returns are uncorrelated with current or past returns. Hence for traded assets the price should follow something more similar to a GBM than an O-U process. However, many financial metrics are not prices; for example interest rates or volatility. O-U processes may describe these processes better than GBM.

A simple (and simplistic) heuristic is: given a price, model with GBM (at least for a first approximation). Given a metric, model with O-U (at least for a first approximation).

• Ok, thanks! If I understand well, the main difference is mean-reversion (for O-U)? Something else: what would you use for FOREX? GBM or O-U? Ex: EUR/USD – Basj Jan 22 '16 at 11:45
• For nominal exchange rates GBM. For real exchange rates [i.e purchasing power adjusted] some say there is long run mean reversion, hence O-U could be used. But keep in mind the reversion is slow (months or years). – noob2 Jan 22 '16 at 15:12

the answer is simple: look at key differences between these two models. GBM is diffusion, OU is mean-reversion

• OU also has a diffusion term. – SmallChess Jan 23 '16 at 13:28