Machine learning Gaussian short rate

Dissertação para obtenção do Grau de Doutor em Estatística e Gestão do Risco

The main theme of this thesis is the calibration of a short rate model under the risk neutral measure. The problem of calibrating short rate models arises as most of the popular models have the drawback of not fitting prices observed in the market, in particular, those of the zero coupon bonds that define the current term structure of interest rates. This thesis proposes a risk neutral Gaussian short rate model based on Gaussian processes for machine learning regression using the Vasicek short rate model as prior. The proposed model fits not only the prices that define the current term structure observed in the market but also all past prices. The calibration is done using market observed zero coupon bond prices, exclusively. No other sources of information are needed. This thesis has two parts. The first part contains a set of self-contained finished papers, one already published, another accepted for publication and the others submitted for publication. The second part contains a set of self-contained unsubmitted papers. Although the fundamental work on papers in part two is finished as well, there are some extra work we want to include before submitting them for publication. Part I: - Machine learning Vasicek model calibration with Gaussian processes In this paper we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. We stress that the only prices needed for calibration are market observed zero coupon bond prices and that the parameters are directly obtained in the arbitrage free risk neutral measure. - One Factor Machine Learning Gaussian Short Rate In this paper we model the short rate, under the risk neutral measure, as a Gaussian process, conditioned on market observed zero coupon bonds log prices. The model is based on Gaussian processes for machine learning, using a single Vasicek factor as prior. All model parameters are learned directly under the risk neutral measure,using zero coupon bonds log prices only. The model supports observations of zero coupon bounds with distinct maturities limited to one observation per time instant. All the supported observations are automatically fitted.

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