1
$\begingroup$

I'm looking for some reference on how to calibrate a non-mean-reverting Ornstein-Uhlenbeck process to historical data using MLE or OLS. The model has the following SDE:

$d\lambda(t)=a\lambda(t)dt+\sigma dW(t)$

with $a>0$ and $\sigma \geq 0$.

Hints on how to adapt the procedure from a mean-reverting OU may be useful too.

$\endgroup$
  • $\begingroup$ suppose the prices of an asset over the past 5 weeks are 10, 15, 12, 20, 22. Then ln(15/10) ... ln(22/10) gives the terms 0.405, -0.223, 0.511, 0.095. You could estimate sigma as the standard deviation of these terms, 0.3311. The drift term could be estimated as the mean of these terms + 0.5 * sigma^2, 0.197 + 0.5 * 0.3311^2 = 0.2518. Not sure if that's what your looking for? $\endgroup$ – nathanesau Jul 8 '15 at 6:45
3
$\begingroup$

EDIT: My reasoning below seems to be wrong. The process as you write it tends to infinity if $a$ is big enough and positive and if $\lambda_0$ is positive. I would not call this process non-meanreverting OU. It is just an Ito process of a simple form. If we remove the stochastic part then we get $$ d\lambda_t = a \lambda_t dt $$ with the solution (if $\lambda_0>0$) $\lambda_t = \lambda_0 \exp(a t)$ which for $a>0$ just grows exponentially. If look at the whole thing then we add a stochastic disturbance at each time step of size $\sigma dB_t$. Thinking about it this way I think that the process above does not have too much in common with an OU-process.

I delete my previous answer.

Concerning the estimation: If you have a process that you have observed on a time grid with width $ \Delta t$ then a discretization of your SDE could look like this: $$ \lambda(t + \Delta t) - \lambda(t) = \theta \lambda(t) \Delta t + \sigma \sqrt{\Delta t} \epsilon_i $$ where $\epsilon_i$ is standard normal. Thus a regression of $\lambda(t + \Delta t) - \lambda(t)$ on $\lambda(t)$ gives you $\theta \Delta t$. The volatility of the residuals gives you an estimate of $\sigma \sqrt{\Delta t}$. Dividing these quantities by the grid width (resp its square-root) gives you the parameters.

Look at a similar question here.

$\endgroup$
  • $\begingroup$ That's a great way to look at it. I'm reading the following paper: belgianactuarialbulletin.be/articles/vol08/02-Luciano.pdf. They call it non-mean-reverting OU. What about estimating the parameters using MLE or OLS? I tried the MLE algorithm provided in Iacus - Simulation adn Inference for Stochastic Differential Equations, using mean and variance of the non-mean-reverting OU, but the numerical optimizer doesn't work properly. $\endgroup$ – Egodym Jul 8 '15 at 14:18
  • $\begingroup$ I have edited the answer. Try it with simulated data. $\endgroup$ – Ric Jul 8 '15 at 14:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.