Add multivariate Vasicek model example

[SamDuffield]

It would be awesome if we can use thermox.log_prob and gradient ascent to fit a multivariate Vasicek model to some real data

[KaelanDt]

At the end of this notebook there is an example for bond pricing with the Vasicek model https://github.com/cantaro86/Financial-Models-Numerical-Methods/blob/afd0e5340502f0056aec00b594c64928b34f4bcb/6.1%20Ornstein-Uhlenbeck%20process%20and%20applications.ipynb

[SamDuffield]

At the end of this notebook there is an example for bond pricing with the Vasicek model https://github.com/cantaro86/Financial-Models-Numerical-Methods/blob/afd0e5340502f0056aec00b594c64928b34f4bcb/6.1%20Ornstein-Uhlenbeck%20process%20and%20applications.ipynb

This seems to be univariate right? But yeah might be a good resource for inspiration for extending to multivariate case

[zauberresonator]

At the end of this notebook there is an example for bond pricing with the Vasicek model https://github.com/cantaro86/Financial-Models-Numerical-Methods/blob/afd0e5340502f0056aec00b594c64928b34f4bcb/6.1%20Ornstein-Uhlenbeck%20process%20and%20applications.ipynb

This seems to be univariate right? But yeah might be a good resource for inspiration for extending to multivariate case

It will be kind of messy if you are trying to obtain some substantial amount of real bond data without access to certain databases (e.g., WRDS (TRACE) database). That said, you can check out the datasets here at Open Source Bond Asset Pricing - https://openbondassetpricing.com/data/ - and I recommend skimming through some Fabozzi to understand what the variables mean in the data files.

[zauberresonator]

At the end of this notebook there is an example for bond pricing with the Vasicek model https://github.com/cantaro86/Financial-Models-Numerical-Methods/blob/afd0e5340502f0056aec00b594c64928b34f4bcb/6.1%20Ornstein-Uhlenbeck%20process%20and%20applications.ipynb

This seems to be univariate right? But yeah might be a good resource for inspiration for extending to multivariate case

What exactly do you mean by a multivariate extension? There is only one short rate modelled in a Vasicek model - presumably you might be thinking of modelling interest rates in different economies (e.g., G7), but then these rates may be highly correlated, so it may be preferable to model and simulate in a more suitable basis where the transformed interest rates are not highly correlated and then transform back to the original interest rates in the end.

[SamDuffield]

@zauberresonator thanks for your insight! You clearly are coming from a more financially applied viewpoint than us which is great!

What exactly do you mean by a multivariate extension?

I was thinking that yeah we could have multiple interest rates/economies within the same model and as you say they might be correlated but these correlations would be learnt by having a higher dimensional Vasicek/OU model that includes dense covariances and drift dependencies. I.e. learning numerically how correlated the interest rates are.

[zauberresonator]

@zauberresonator thanks for your insight! You clearly are coming from a more financially applied viewpoint than us which is great!

What exactly do you mean by a multivariate extension?

I was thinking that yeah we could have multiple interest rates/economies within the same model and as you say they might be correlated but these correlations would be learnt by having a higher dimensional Vasicek/OU model that includes dense covariances and drift dependencies. I.e. learning numerically how correlated the interest rates are.

Wow, that is cool - I have studied the relevant subjects in the context of physics, statistics, and finance in uni before. I will try this out on my own over the holidays, but you can ping me at [email protected] if I can be of help offline. There are a few more things I will note if you are trying this out:
(i) Covariance matrix: typical (high-dimensional) covariance matrix estimation often assumes some form of low-rank structure (revealed through various rank-revealing matrix factorisations) and/or applies some form of regularisation/shrinkage to only use the "important" components; this can be visualised by plotting the eigenvalues (from largest to smallest) for naive and shrunk sample covariance matrices
(ii) Precision matrix: assuming multivariate Gaussian distribution, entry (i,j) of the precision matrix is related to the partial correlation of $x_i$ and $x_j$ conditioning on the other $x_k$ variables and is precisely zero when $x_i$ and $x_j$ are conditionally independent (given the other variables); similar high-dimensional concerns apply to the sample estimate precision matrix
(iii) RMT: naturally, there is all the beautiful results from random matrix theory going back to Wigner, Dyson, and Mehta, but I will be careful before applying those as sanity checks because the relevant limit in this context is $T \rightarrow \infty$, $N \rightarrow \infty$, and $q=\frac{N}{T}=O(1)$