  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.
