Essays on nonlinear filtering with applications in finance
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Abstract
In this dissertation, I discuss the nonlinear filtering problem and its applications in finance. In the first chapter, I present a new filtering approach for nonlinear and non- Gaussian state space models. This approach builds on the well-established Kalman filter, featuring a state-dependent least-square linearization of the nonlinear function and a Gaussian-mixture approximation to the error distribution, and it applies the quasi-Monte Carlo method for numerical integration during the computation. The theoretical analysis shows that when the model is Gaussian, the filtering distribution based on the proposed approach can capture the true first two moments of the state variable; when the model is non-Gaussian, the filtering distribution can always be represented by a Gaussian mixture. This study also provides an analysis of the stability of this new filtering approach. In addition, I propose two methods to estimate the unknown parameters of the model. The first is an off-line likelihood-based method, and the second is an on-line dual estimation method. I also establish the consistency of the proposed quasi-maximum likelihood estimator under general conditions. To illustrate the proposed approach, I discuss several numerical examples using simulated data and compare my approach with other existing methods. I find that the proposed approach can outperform these methods in terms of speed and accuracy.
The second chapter examines pairs trading using a general state space model framework. I model the spread between the prices of two assets as an unobservable state variable. I show how to use the filtered spread to carry out statistical arbitrage. I also propose a new trading strategy and present a Monte Carlo-based approach to select the optimal trading rule.
The third chapter, coauthored with Li Chen, presents a new approximation scheme for the price and exercise policy of American options. The scheme is based on Hermite polynomial expansions of the transition density of the underlying asset dynamics and the early exercise premium representation of the American option price. The proposed approximations of the price and optimal exercise boundary are shown to be convergent to the true ones.