Abstract
Estimating covariances between financial assets plays an important role in risk management. In practice, when the sample size is small compared to the number of variables, the empirical estimate is known to be very unstable. Here, we propose a novel covariance estimator based on the Gaussian Process Latent Variable Model (GP-LVM). Our estimator can be considered as a non-linear extension of standard factor models with readily interpretable parameters reminiscent of market betas. Furthermore, our Bayesian treatment naturally shrinks the sample covariance matrix towards a more structured matrix given by the prior and thereby systematically reduces estimation errors. Finally, we discuss some financial applications of the GP-LVM.
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Notes
- 1.
The evidence \(\log p(\varvec{Y})\) is also referred to as log marginal likelihood in the literature. The term marginal likelihood is already used for \(p(\varvec{Y}|\varvec{X})\) in this paper. Therefore, we will refer to \(\log p(\varvec{Y})\) as the evidence.
- 2.
To stay consistent with the financial literature, we denote the return matrix with lower case \(\varvec{r}\).
- 3.
Because of the context, from now on we will use \(\varvec{\beta }\) for the latent space instead of \(\varvec{x}\).
- 4.
We are aware that this introduces survivorship bias. Thus, in some applications our results might be overly optimistic. Nevertheless, we expect relative comparisons between different models to be meaningful.
- 5.
- 6.
Here, we have used the implementation in the Python toolbox scikit-learn [20].
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The authors thank Dr. h.c. Maucher for funding their positions.
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Nirwan, R.S., Bertschinger, N. (2020). Applications of Gaussian Process Latent Variable Models in Finance. In: Bi, Y., Bhatia, R., Kapoor, S. (eds) Intelligent Systems and Applications. IntelliSys 2019. Advances in Intelligent Systems and Computing, vol 1038. Springer, Cham. https://doi.org/10.1007/978-3-030-29513-4_87
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