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 model. Many financial problems require the estimation of covariance matrices between given assets. This may be useful to optimize one's portfolio, i.e.: maximize the portfolio returns w T r and/or minimize the volatility √ w T Kw. Indeed, Markowitz received a Noble Price in economics for his treatment of modern portfolio theory BID9 . In practice, estimating historical returns and high-dimensional covariance matrices is challenging and often times equally weighted portfolio outperforms the portfolio constructed from sample estimates BID6 . The estimation of covariance matrices is especially hard, when the number of assets is large compared to the number of observations. Sample estimations in those cases are very unstable or can even become singular. To cope with this problem, a wide range of estimators, e.g. factor models such as the single-index model BID16 or shrinkage estimators BID8 , have been developed and employed in portfolio optimization.With todays machine learning techniques we can even further improve those estimates. Machine learning has already arrived in finance. BID10 trained an agent via reinforcement learning to optimally execute trades. BID4 forecast asset prices with neural networks and BID1 with Gaussian Processes. Recently, BID5 made an ansatz to optimally allocate portfolios using deep autoencoders. BID21 used Gaussian Processes to build volatility models and BID20 to estimate time varying covariance matrices. Bayesian machine learning methods are used more and more in this domain. The fact, that in Bayesian framework parameters are not treated as true values, but as random variables, accounts for estimation uncertainties and can even alleviate the unwanted impacts of outliers. Furthermore, one can easily incorporate additional information and/or personal views by selecting suitable priors.In this paper, we propose a Bayesian covariance estimator based on the Gaussian Process Latent Variable Model (GP-LVM) BID7 , which can be considered as a non-linear extension of standard factor models with readily interpretable parameters reminiscent of market betas. Our Bayesian treatment naturally shrinks the sample covariance matrix (which maximizes the likelihood function) towards a more structured matrix given by the prior and thereby systematically reduces estimation errors. We evaluated our model on the stocks of S&P500 and found significant improvements in terms of model fit compared to classical linear models. Furthermore we suggest some financial applications, where Gaussian Processes can be used as well. That includes portfolio allocation, price prediction for less frequently traded stocks and non-linear clustering of stocks into their sub-sectors.In section 2 we begin with an introduction to the Bayesian non-parametric Gaussian Processes and discuss the associated requirements for learning. Section 3 introduces the financial background needed for portfolio optimization and how to relate it to Gaussian Processes. In section 4 we conduct experiments on covariance matrix estimations and discuss the results. We conclude in section 5. We applied the Gaussian Process Latent Variable Model (GP-LVM) to estimate the covariance matrix between different assets, given their time series. We then showed how the GP-LVM can be seen as a non-linear extension to the CAPM with latent factors. Based on the R 2 -score and the ELBO, we concluded, that for fixed latent space dimension Q, every non-linear kernel can capture more structure than the linear one.The estimated covariance matrix helps us to build a minimal risk portfolio according to Markowitz Portfolio theory. We evaluated the performance of different models on the S&P500 from year 2008 to 2018. Again, non-linear kernels had lower risk in the suggested portfolio and higher Sharpe ratios than the linear kernel and the baseline measures. Furthermore, we showed how to use the GP-LVM to fill in missing prices of less frequently traded assets and we discussed the role of the latent positions of the assets. In the future, one could also put a Gaussian Process on the latent positions and allow them to vary in time, which would lead to a time-dependent covariance matrix.
Covariance matrix estimation of financial assets with Gaussian Process Latent Variable Models Illustrates how the Gaussian Process Latent Variable Model (GP-LVM) can replace classical linear factor models for the estimation of covariance matrices in portfolio optimization problems. This paper uses standard GPLVMs to model the covariance structure and a latent space representation of S&P500 financial time series, to optimize portfolios and predict missing values. This paper proposes to use a GPLVM to model financial returns