Estimation of Financial Models Using Moment Conditions Defined on Frequency Domain

Abstract

This dissertations presents the estimation methods of financial models for which the density function is not known in closed form, but the characteristic function (or Laplace transform) is available in the analytical form. In this case the estimation is done via Generalized Method of Moments, where moment conditions are defined on frequency domain. The dissertation consists of three chapters. The first chapter develops a method of selection of finite set of moment conditions out of infinite number of possibilities using proposed optimality criterion. The second chapter describes the estimation of parametric asset price models using finite set of moment conditions based on characteristic function. The final chapter proposes a method for estimating parametric models for stochastic volatility using moment conditions based on the integrated Laplace transforms. Chapter 2 develops a new estimator for the case where the moment function is the difference between model-implied and data-implied characteristic functions. There is an optimal GMM estimator that attains the Cramer-Rao lower bound and uses all continuum of moment conditions. However, the implementation of continuum moments GMM is not practical. I develop a practical and consistent procedure to select small finite subset of moment conditions that yields a nearly efficient estimator. The moment selection algorithm works by approximating the span of the continuum of moment conditions by an optimal finite subset. The method involves a metric to evaluate how close the asymptotic variance is to the maximum likelihood estimator asymptotic variance in relative terms. The Monte Carlo application for a jump-diffusion model indicates that it is enough to use 9 moment conditions to have almost efficient estimator. Chapter 3 is co-authored with Professor George Tauchen. We present parametric estimation of models for stock returns by describing price dynamic as the sum of two independent L'{e}vy components. The increments (moves) are viewed as discrete-time log price changes that follow an infinitely divisible distribution, i.e. stationary and independent price changes (zero drift) that follow a L'evy-type distribution. We explore empirical plausibility of two parametric models: Jump-Diffusion (C-J) and pure jump model (TS-J). The first component of each model describes the dynamics of small frequent moves and is modeled by Brownian motion in C-J model and by tempered stable L'evy process in TS-J model. The second component is responsible for big rare moves in asset prices and is modeled by compound Poisson process in both models. Using high frequency data on 13 stocks of different market capitalization for 2006-2008 sample period we find that C-J model performs well only for large cap stocks, while medium cap stock dynamics are captured by TS-J model.Chapter 4 is co-authored with Professor George Tauchen and Professor Viktor Todorov. We propose analytically tractable way to estimate parametric models for unobserved stochastic volatility. The estimation works my matching moments of the integrated joint Laplace Transform of volatility with those implied by parametric volatility model. We use so-called Realized Laplace Transform of volatility to receive a model-free and jump-robust estimate of integrated empirical Laplace transform of the unobservable volatility. Monte Carlo reveals the proposed estimation method is very efficient. Empirical application using the method reveals that market volatility has both persistent and transient volatility spikes and both of them are sufficiently volatile which necessitates rich jump specifications to capture the observed patterns in stochastic volatility.