2
$\begingroup$

Does anyone know how to estimate $A$, $\sigma_1$,$\sigma_2$ from the following system?

$$dx = \mu_t x dt + \sigma_1 x dB_x$$

$$d\mu = A(\bar\mu - \mu) dt + \sigma_2 dB_\mu$$

Variation in $x$ could be either attributed to variation in $\mu$, or variation in $dB_x$, right?

Suppose I know $\bar \mu$, but need to estimate all the rest of the parameters.

Improve this question
$\endgroup$
4
  • $\begingroup$ Please have a look at what I did to make sure your math formatting appears properly next time. $\endgroup$
    – SRKX
    Commented Mar 5, 2015 at 1:49
  • $\begingroup$ Also, it would be interesting to see where you got this model from (or did you come up with it)? Didn't this source provide any clues on how to calibrate? What have you tried so far? $\endgroup$
    – SRKX
    Commented Mar 5, 2015 at 2:04
  • 1
    $\begingroup$ What is the assume correlation between $B_\mu$ and $B_x$? $\endgroup$
    – SRKX
    Commented Mar 5, 2015 at 2:20
  • $\begingroup$ which kind of estimation of the model do you have in mind? estimation to time-series data? option prices? $\endgroup$
    – Gabriele Pompa
    Commented Mar 5, 2015 at 11:18

3 Answers 3

Reset to default
3
$\begingroup$

I would say

  1. Take log of first equation to get rid of dependence on $x_t$
  2. Apply Kalman filter equations to estimate parameters

I believe Conrad and Kaul (1988) J of Business do exactly what you describe.

Improve this answer
$\endgroup$
1
$\begingroup$

It depends on the use of your model as pointed out in the comments. If a discretized version is sufficient then state space models could be a solution.

You can check out the free online textbook by Athana­sopou­los and Hyndman. State space model describe time series in terms of level/trend (and seasonality) on an additive or multiplicative way. There are nice procedure and packages to estimate and forecast such models.

Improve this answer
$\endgroup$
1
$\begingroup$

Thank you guys. Sorry for the late reply, I just solved it in matlab using maximum likelihood estimation. Turns out that all we need to do is to specify a state space model, then estimate the coefficient using MLE. The linearity and normality here makes things pretty simple.

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ What would actually be awesome is if could post your code here! I'm sure it would definitely help other people in the future! $\endgroup$
    – SRKX
    Commented Mar 12, 2015 at 7:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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