Serving the Quantitative Finance Community

 
User avatar
EdisonCruise
Topic Author
Posts: 54
Joined: September 15th, 2012, 4:22 am

How to calibrate this stochastic model?

June 28th, 2021, 12:23 pm

I meet some problems when calibrating the stochastic model, given in Eq(3.2) and appendix (E.2) of thesis:
https://tspace.library.utoronto.ca/bits ... thesis.pdf

I give a simplified version here.
$$d\lambda_t=\beta(\theta-\lambda)dt+\eta dM$$
where M is a poisson process.
Suppose \({t_1,t_2,...,t_n}\) is a set of observed times with \({t_n<=T}\). The hazard rate is given by \(\lambda_t=\theta+ \sum_{i=1}^{n} \eta e^{-\beta(t-t_i)} \). The log-likelihood function is 
$$L=-\theta T+\sum_{i=1}^{n} \{{log(\lambda_{t_i})-\eta \frac{1-e^{-\beta(T-t_i)}}{\beta}}\}$$
The above log-likelihood function seems questionable to me, because it seems that \(L\) can be maximized by simply choosing \( \theta\rightarrow 0, \eta\rightarrow +\infty, \beta\rightarrow +\infty \). This is also what happens when I implement it with matlab fmincon. So do I miss anything? Thank you in advance.
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: How to calibrate this stochastic model?

June 28th, 2021, 4:38 pm

This looks like a good account -- might help. 
Not sure if (4.5) there is any different than what you posted (assuming they are the same process)
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: How to calibrate this stochastic model?

June 28th, 2021, 4:56 pm

If they're the same log-likelihood, try simulating the process with round number parameters and use two-sided bounds for each during the MLE. This will prevent runaway parameters. Use a constrained non-linear optimizer that you trust.
 
User avatar
EdisonCruise
Topic Author
Posts: 54
Joined: September 15th, 2012, 4:22 am

Re: How to calibrate this stochastic model?

June 29th, 2021, 3:21 am

Thank you so much. I find the problem. Eq (4.5)  in your reference is clear. 
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: How to calibrate this stochastic model?

June 29th, 2021, 4:43 pm

You're welcome. And from looking at it yesterday, I recall two (small?) differences from (4.5) vs. yours: 
a different limit on the [$]\lambda_t[$] sum:  n-1, and the requirement that [$]t_n = T[$].