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Archive for November 20th, 2013

Estimation of Exponential Ornstein-Uhlenbeck process

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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.

Written by fxpaul

November 20, 2013 at 13:04

Posted in Uncategorized