Gaussian Process Regression for Volatility and Covariance Estimation of Noisy Observations

Abstract

Estimation volatilities and covariances of asset returns is a central task for many practitioners in the financial industry. For example, estimating volatility is a necessary prerequisite for options pricing in the Black-Scholes world while knowing variances and covariances between returns for a family of risky assets is a necessary prerequisite for optimal portfolio construction. However, in the presence of noisy price observations, estimation of these quantities is nontrivial. Though statistical methods exist, they require a plethora of assumptions that may lead to a misspecified model. Machine learning techniques, on the other hand, have found great success in finding structure in noisy and data-intensive environments with relatively few model assumptions. As such, this thesis applies Gaussian process regression, a robust and flexible machine learning model, to solve this task of estimating volatilities and covariances in both the noiseless and noisy settings. To evaluate our models, we apply them to the S&P 500 market index and compare our results to known metrics of market volatility. Additionally, we benchmark our methods by evaluating the performance of optimal portfolios constructed using volatilities and covariances outputted by our models.