# FxPaul

Math in finance or vice versa

## Estimation of Exponential Ornstein-Uhlenbeck process

In the previous article about process calibration we derived the following updating formula:
$P_t - P_{t-1} = \left(\theta(\mu - \ln P_{t-1}) + \frac{1}{2} \sigma^2 \right) P_{t-1} \Delta t + \sigma P_{t-1} \sqrt{\Delta t} W_{t-1}$
Now we can rearrange equation’s parts as follows:
$\frac{P_t - P_{t-1}}{P_{t-1}} = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t - \theta\Delta t \ln P_{t-1} + \sigma \sqrt{\Delta t} W_{t-1}$
Thus we can equate it against simple regression formula:
$y = a + bx + \epsilon$

Therefore, we obtain:
$y = \frac{P_t - P_{t-1}}{P_{t-1}}$
$x = \ln P_{t-1}$
$b = - \theta\Delta t$
$a = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t$
$\epsilon = \sigma \sqrt{\Delta t} W_{t-1}$
And this gives us the following OLS estimates:
$\theta = - \frac{b}{\Delta t}$
$a = - b \mu + \frac{1}{2} \sigma^2 \Delta t$
As it was in the previous articles:
$\sigma^2 = \frac{\sigma^2_{\epsilon}}{\Delta t}$
$a = \frac{1}{2} \sigma^2_{\epsilon} - b \mu$
And finally the estimation of $\mu$ is:
$\mu = \frac{\frac{1}{2} \sigma^2_{\epsilon} - a}{b}$

Thus we obtained the estimation of exponential Ornstein-Uhlenbeck process.