Stochastic Approaches to Asset Price Analysis

In this project, we propose to explore the Kalman filter's performance for estimating asset prices. We begin by introducing a stochastic mean-reverting processes, the Ornstein-Uhlenbeck (OU) model. After this we discuss the Kalman filter in detail, and its application with this model. After a demonstration of the Kalman filter on a simulated OU process and a discussion of maximum likelihood estimation (MLE) for estimating model parameters, we apply the Kalman filter with the OU process and trailing parameter estimation to real stock market data. We finish by proposing a simple day-trading algorithm using the Kalman filter with the OU process and backtest its performance using Apple's stock price. We then move to the Heston model, a combination of Geometric Brownian Motion and the OU process. Maximum likelihood estimation is commonly used for Heston model parameter estimation, which results in very complex forms. Here we propose an alternative but easier way of parameter estimation, called the method of moments (MOM). After the derivation of these estimators, we again apply this method to real stock data to assess its performance.