Choosing Geometric Brownian Motion or Ornstein–Uhlenbeck for Financial Data

# Geometric brownian motion vs. Ornstein Uhlenbeck


# 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 ?

## Answer by achirikhin (score 4)

https://quant.stackexchange.com/a/79324

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.

- 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. $Y_t = \log{X_t}$.

- 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.

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

- 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)

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

## Answer by user9403 (score 3)

https://quant.stackexchange.com/a/22866

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).

## Answer by mxzzzzz (score -1)

https://quant.stackexchange.com/a/22870

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