Parametric Estimation of the Heston Model under the Indirect Observability Framework

Estimating parameters in a given stochastic model from a discrete dataset has wide applications in various scientific studies. However, it is also common that the available data are not generated from the stochastic model under investigation, but come from some other sources. For instance, realized volatility is often used to approximate the actual volatility process, since the actual volatility cannot be obtained directly. Therefore, inherent differences between the data and the model may naturally lead to the question about the consistency and robustness of statistical estimation of parameters in the chosen stochastic model.

Three parts are presented in this dissertation. In part I (Chapter 2) of this dissertation we show that the Method of Moments can be used to derive consistent and robust estimators when it is applied to estimate parameters in a stationary non-Gaussian stochastic process with indirect observations. More precisely, we define the unobservable limiting process and the observable approximating process, state necessary hypotheses about both processes under the \emph{Indirect observation framework,} and prove the $L2$ convergence between the empirical covariance function (based on the observed data which are sampled from the approximating process) and the actual covariance function of the limiting process. Hence, the consistent and robust parameter estimators based on the Method of Moments are obtained.

In part II (Chapters 3 and 4), we introduce the Heston model and the associated realized variance process as an application of the general theorem in the first part of the dissertation. In order to apply the general theorem, we show that the variance process from the Heston model satisfies all the hypotheses of the limiting process. At the same time, we also show that the realized variance process uniformly converges to the actual variance process in $L4.$ Therefore, the realized variance process also satisfies the hypotheses of the approximating process under the \emph{Indirect observation framework.} Moreover, we derive an optimal sampling scheme to construct the realized variance process for approximating the actual variance process such that the requirement of accuracy is balanced with data restrictions or computational costs. The part II of this dissertation provides theoretical support for the existence of consistent and robust parameter estimators for the variance process in the Heston model when the realized variance data is used.

In the last part of this dissertation, we perform numerical simulation of the Heston model to validate our bound estimator obtained theoretically in Part II. First, we numerically verify the uniform convergence between the actual volatility and the realized volatility in $L2$ and $L4,$ and illustrate the optimal sampling criteria for constructing the realized volatility data. Then the $L2$ accuracy of parameter estimators based on the realized volatility dataset is confirmed. Our numerical evidences strongly support the theoretical results obtained in Part II and suggest a scaling law for the $L2$ convergence of parameter estimators.