Volatility estimation with functional gradient descent for very high-dimensional financial time series

Abstract

We propose a functional gradient descent algorithm (FGD) for estimating volatility and conditional covariances (given the past) for very high-dimensional financial time series of asset price returns. FGD is a kind of hybrid of nonparametric statistical function estimation and numerical optimization. Our FGD algorithm is computationally feasible in multivariate problems with dozens up to thousands of individual return series. Moreover, we demonstrate on some synthetic and real data-sets with dimensions up to 100, that it yields significantly, much better predictions than more classical approaches such as a constant conditional correlation GARCH-type model. Since our FGD algorithm is constructed from a generic algorithm, the technique can be easily adapted to other problems of learning in very high dimensions.

Semi-parametric forecasts of the implied volatility surface using regression trees

“…Friedman et al 2000;Friedman 2001). Audrino and Bühlmann (2003) already successfully applied FGD to estimate volatility in high-dimensional GARCH models, and Audrino et al (2005) used it to model interest rates.…”

“…5 This machine learning technique has shown its power in improving volatility forecasts in high-dimensional GARCH models (see Audrino and Bühlmann 2003). In another case, FGDbased filtered historical simulation was conducted to compute reliable out-of-sample yield curve scenarios and confidence intervals (see Audrino and Trojani 2007).…”

An Introduction to Boosting and Leveraging

“…Some further applications to Pose Invariant Face Recognition [94], Lung Cancer Cell Identification [200] and Volatility Estimation for Financial Time Series [7] have also been developed.…”

A Multivariate FGD Technique to Improve VaR Computation in Equity Markets

“…Choosing reasonable starting functions (for example estimated by a very simple multivariate GARCH-type model), FGD tries to improve, often successfully, those components where the initial predictions are poorest. Clearly, as Audrino and Bühlmann (2003) have already shown, we can not expect to learn in all d dimensions when increasing d and keeping a fixed sample size. However, although the gain on average will generally decrease with the number of fitted assets, FGD still improves the worst cases.…”

“…FGD is a recent technique from the area of machine learning introduced to solve the classification problem (Mason et al, 1999;Breiman, 1999;Friedman et al, 2000;Friedman, 2001). The FGD algorithm that we propose is the same as in Audrino and Bühlmann (2003), who have studied the statistical performance of FGD in the financial field. It is very general and can be further adapted to solve other multivariate problems dealing with high-dimensional (volatility) function estimation, such as asset allocation problems, involving the allocation of assets among several stocks whose returns are correlated, or risk management for large global trading portfolios with time-dependent weights.…”