Statistical estimation of jump-diffusion models via optimal thresholding

Jeffrey Nisen, Purdue University

Abstract

Accurate volatility estimation is of fundamental importance for many problems in asset pricing, risk management, and portfolio selection contexts. By studying the first and second order moment asymptotics for some commonly used volatility estimators, we provide necessary and sufficient conditions for their mean-squared convergence, hence, sharpening previous works which have mostly focused on providing only sufficient conditions. Furthermore, we reveal the precise connection between jump detection and volatility estimation. Concretely, we find that within the class of threshold based estimators, there are three main sources of estimation error, namely, false positive jump misclassification error, false negative jump misclassification error, and error due to the intrinsic natural variability of the estimators. Inspired by this fundamental error decomposition, we proceed to introduce a novel and well-posed optimization problem that aims at selecting estimators which minimize the first two sources of error. We analyze this problem for a certain class of jump diffusion processes with possibly time varying drift, volatility, and jump intensity, and demonstrate the existence and uniqueness of the "optimal" threshold estimator. Moreover, we provide both an asymptotic and a fixed point characterization of the optimal thresholding level for the threshold estimators. Building on these characterizations, we then propose novel estimation algorithms, which allow for feasible implementations of the optimal threshold estimators. An extensive Monte Carlo study demonstrates the improved finite sample performance of the new estimators relative to many commonly used alternatives. Our results open up new and interesting directions for statistical estimation of stochastic processes with jumps.

Degree

Ph.D.

Advisors

Figueroa-Lopez, Purdue University.

Subject Area

Statistics

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