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EdisonCruise
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How to estimate this stochastic model?

May 9th, 2022, 9:53 am

$$dS_t=\alpha_tdt+\sigma_1dW^1_t$$
$$d\alpha_tdt=-\xi\alpha_tdt+\sigma_2dW^2_t$$
where \(S_t\) is the observable price; \(\alpha_t\) is the unobservable trend; \(dW^1_t\) and \(dW^2_t\) are two Brownian motions.
I think \(S_t\) can be sampled with regular discrete time, so that this model can be formulated as a state space model and be estimated by MATLAB.
https://www.mathworks.com/help/econ/estimate-a-time-invariant-state-space-model.html
However, MATLAB uses general Kalman filter to do this, which takes lots of computational effort. Is there any more efficient algorithm to estimate this specific model?
 
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Marsden
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Re: How to estimate this stochastic model?

May 9th, 2022, 10:30 am

Is the left hand of your second equation correct? Did you mean to include dt in it?
 
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EdisonCruise
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Re: How to estimate this stochastic model?

May 9th, 2022, 11:04 am

Is the left hand of your second equation correct? Did you mean to include dt in it?
sorry, that's a typo. You are right. How can I edit the equation after submission?
 
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Marsden
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Re: How to estimate this stochastic model?

May 9th, 2022, 11:29 am

I don't think you can, unfortunately.

Easy enough to work around ... not that I'm thinking I'll have an answer for you any time soon. ;-)
 
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Marsden
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Re: How to estimate this stochastic model?

May 9th, 2022, 11:53 am

α essentially has exponential growth (or shrinkage, if you want the coefficient of αtdt to be negative ...) plus noise.

If α is shrinking, the noise starts to dominate. So S would have a variable drift with a starting value that fades in significance over time (did you mean for either of the terms on the right hand of your first equation to be multiplied by St? Otherwise, is it very different from plain vanilla Brownian motion with some starting parameters that disappear over time?)
 
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bearish
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Re: How to estimate this stochastic model?

May 9th, 2022, 1:12 pm

If the context for this equation (mutatis mutandum) is the price of a stock, or some other asset with a similar volatility, your task is essentially hopeless. It takes years (not to say decades) of data to get a decent estimate of a constant drift, and estimating the parameters of a stochastic drift is only making it worse.
 
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Alan
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Re: How to estimate this stochastic model?

May 9th, 2022, 6:13 pm

It's a terrible model for a traded security price for various reasons and, as a bearish has noted, drifts are very difficult to estimate.

Nevertheless, the direct answer to your question is that parameter estimation can be done approximately using a Hidden Markov model approach. There's discussion in my 'Option Valuation under Stochastic Volatility II" for the case of volatility being the latent (unobserved) variable. Your case with the drift parameter being the latent variable would be similar. Basically, you would need to discretize [$]\alpha_t[$] to a (finite) discrete state space and then matrix arithmetic can produce an (approximate) log-likelihood function, which you then maximize over your free parameters.

There's a fairly steep learning curve involved. It may not be worth the effort unless you want to learn the general technique.  If the Kalman filter can handle your model, probably better to use that.
 
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EdisonCruise
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Re: How to estimate this stochastic model?

May 10th, 2022, 3:25 am

Thank you so much for your suggestions. Actually this model appears frequently in Prof. Alvaro Cartea’s high frequency trading paper. I need to figure out a way to calculate \(\alpha_t\) and estimate parameters in real trading efficiently. Maybe I can some approximations of this model by following the idea in “An analytic approximation of the likelihood function for the Heston model volatility estimation problem”