# 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. Commented Jan 22, 2016 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
Commented Jan 22, 2016 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. Commented Jan 22, 2016 at 16:06

A more abstract yet simple way of looking at this may help.

Consider a generic diffusion

$$dY = (a_t - b_t Y_t) dt + \sigma_t dW_t$$,

where $$Y_t$$ is either the modelled quantity itself or $$Y_t = \log{X_t}$$ for some other quantity $$X_t>0$$.

This equation generalizes all your cases and different features of the equations are either used or eliminated depending on what you are modelling.

1. If you are ultimately modelling a non-negative quantity, like generic "index": asset like, e.g. equty, fx, inflation index, (credit) hazard rate or any other non-negative yield-like quantity (spread), then you merely model the logarithm, i.e. $$Y_t = \log{X_t}$$.

2. When you don't need "mean reversion", then you put $$b_t=0$$. This is usually done for equity modelling, as in standard BS, but even for equity it depends on the modelling horizon. For certain periods in history, the hypothesis that equity returns are mean reverting cannot be rejected. The non-equities case is that of, say, exponential HW or BK models for the yeild-like quantities, real rates or hazard rates, where you need to have both mean-reversion and positivity.

3. $$a_t$$ usually has be made time dependent to account for various "spot curves", e.g. forward curve for rates or funding (repo) curve for equity

4. Volatility term $$\sigma_t$$ can be as rich as you need, constant, time dependent, local, stochastic, local stochastic, regime switching...

Note that in call cases the quantity $$Y_t$$ is stochastic, i.e. unpredictable, but in the case of $$b_t \neq 0$$ distribution of its increment is conditional on the current value. But it is as stochastic otherwise, as in the case when $$b_t = 0$$.

Just for completeness, the easiest discrete time version of the above, which is usually used in historical "P measure" modelling is, of course, AR(1)

$$Y_n = A + B Y_{n-1} + \eta_n$$

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
Commented Jan 22, 2016 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). Commented Jan 22, 2016 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. Commented Jan 23, 2016 at 13:28