Limits of Ornstein-Uhlenbeck Models for Exchange Rates
Summary
The discussion evaluates fitting an Ornstein-Uhlenbeck process to daily EUR/USD observations using maximum likelihood. The estimate of mean reversion is close to zero, while the reported stationarity test does not reject a non-stationary series. Since the standard OU process assumes stationary, mean-reverting behavior, weak estimated reversion can leave simulations behaving much like Brownian motion and poorly representing the observed exchange-rate path.
Responses suggest that exchange rates may exhibit changing regimes, rare jumps, fat-tailed returns, and time-varying volatility. Alternatives include random walks, autoregressive or ARIMA models, GARCH volatility models, and jump diffusion; a piecewise long-run mean or exponential OU process is also suggested. The discussion notes that an OU process does not ensure positive rates and cautions against using mean reversion as a trading model for FX. These are suggestions rather than a comparative empirical evaluation; the data and calibration code are not examined, so the cause of the weak estimate remains uncertain.
Key ideas
- A standard Ornstein-Uhlenbeck model is intended for stationary, mean-reverting series.
- A near-zero mean-reversion estimate can make simulated paths resemble Brownian motion.
- Exchange-rate data may include jumps, fat tails, and changing volatility that a basic OU model misses.
- Alternative models mentioned include ARIMA, GARCH, jump diffusion, and exponential OU processes.
- Model suitability and calibration quality cannot be assessed without inspecting the data and estimation procedure.
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# Modelling EUR/USD rate with Ornstein-Uhlenbeck model
# Modelling EUR/USD rate with Ornstein-Uhlenbeck model
I have a data set of daily EUR/USD rate for time period 2000-2018.
My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) d t + \sigma d B_t$$ In order to calibrate the model I used MLE (Maximum Likelihood estimation) to obtain estimators for model parameters $\alpha, \theta$ and $\sigma$. Bellow are resulting estimators:
$$\hat{\alpha} = 0.00103 \\ \hat{\theta} = 1.27334 \\ \hat{\sigma} = 0.00647 $$
Whereas estimated value for $\theta$ seems to be pretty reasonable (long-term mean to which the process tends to revert), value for $\alpha$ being close to $0$ does not make much sense to me. As a result the first mean-reverting part of the equation basically drops out and what is left is "weak" (due to small value of $\sigma$) Brownian motion $d X_t = \hat{\sigma} d B_t$. Consequently my simulated process is not modelling the real process at all as can be seen in the picture below:
I am aware of the fact that this problem might be related to lack of stationarity in my data (p-value for ADF test was around 0.6, meaning we did not reject null hypothesis of non-stationarity).
Is this the only reason for poor modelling power of my calibrated model? If so, what are the best ways to tackle the problem of non-stationarity? Are there any other possible reasons for $\alpha$ being close to 0? Also, is O-U model even suitable for modelling exchange rate processes? If not, what models would be more approriate?
## Answer by ricmarchao (score 1)
https://quant.stackexchange.com/a/40504
regarding lack of stationarity: yes, OU process is only viable for stationary processes, and it makes sense that EURUSD data are non-stationary, few rare events can lead to sharp moves: Greece crises, Trump election, etc..the distribution of returns has fat tails and the volatility is heteroskedastic
Most simple financial model is the random walk , AR(1) process, you can also try ARIMA ,GARCH models,Merton's jump diffusion model, the last two take care of the heteroskedasticity of the data
FYI instead of using MLE to estimate the OU parameters you can also use linear regression, the results should be the same:
How can I estimate the Ornstein-Uhlenbeck paramters of some mean reverting data that I have on R?
## Answer by Yuanlin Dong (score 0)
https://quant.stackexchange.com/a/75923
Q: Is this the only reason for poor modelling power of my calibrated model?
My comment: It is hard to tell without taking a close look at the data and your calibration script.
Q: If so, what are the best ways to tackle the problem of non-stationarity?
My comment: One way to tackle the problem of non-stationarity is to model $\theta$ as a piece-wise parameter.
Q: Are there any other possible reasons for 𝛼 being close to 0?
My comment: Hard to tell without looking into the details.
Q: Also, is O-U model even suitable for modelling exchange rate processes? If not, what models would be more approriate?
My comment: I would suggest that you try the exponential O-U process. The fx rate EUR/USD is always positive by its nature, but an O-U process can not guarantee the positiveness. One more thing is that you can not use a mean-reverting process to model the dynamics of FX rates if the intention is to use the model for trading activities.
Good luck!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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