Doctor of Philosophy
Statistics and Actuarial Sciences
The advent of the big data era presents new challenges and opportunities for those managing portfolios, both of assets and of risk exposures, for the financial industry. How to cope with the volume of data to quickly extract actionable information is becoming more crucial than ever before. This information can be used, for example, in pricing various financial products or in calculating risk exposures to meet (ever changing) regulatory requirements.
Stochastic differential equations are often used to model the risk factors in finance. Given the presumption of a functional form for the coefficients of these equations, the required parameters can be calibrated using a large body of statistical techniques which have been developed over the past decades. However, the price to pay for this convenience is the problems that occur if an incorrect functional form is used. To avoid this problem of misspecification, nonparametric methods have recently become important in finance. In order to adequately estimate local structures, large sample sizes are always required and so nonparametric methods are computationally intensive.
This thesis finds new ways to decrease the computational cost of non-parametric methods for estimating stochastic differential equations. Motivated by stochastic approximations, we propose online nonparametric methods to estimate the drift and diffusion terms of typical financial stochastic differential equations. Both stationary and non-stationary processes are considered and this thesis provides asymptotic properties of the estimators. For the stationary case, quadratic convergence, strong consistency, and asymptotic normality of the estimators are established; for the non-stationary case, weak consistency of the estimators is proved.
In addition to numerical examples, we also apply our methods to market risk management. We work from up to date examples based on the most recent Basel Committee documents for a wide range of risk factors from equity, foreign exchange, interest rates, and commodity prices. The advantages and disadvantages of applying our new statistical techniques to these risk management problems are also discussed.
Wang, Xin, "Online Nonparametric Estimation of Stochastic Differential Equations" (2015). Electronic Thesis and Dissertation Repository. 2755.