| The stationary, continuous time, discrete space (SCD) model is developed, and this dissertation highlights some aspects of the SCD model. The SCD model, a class of stationary Markov processes, is useful to investigate phenomenon in financial micro data where information about time in seconds and a trading price for each transaction is available. The SCD model has two components; one component for the inter arrival time, the time of two adjacent transactions, and one component for the transition of a stock price to a new price. The representation theorem for Markov processes assures that the inter arrival time follows the exponential distribution, and the sequence of the stock prices is a Markov chain with some transition matrix. This dissertation suggests three models for the inter arrival time; (i) one parameter model in the exponential distribution, (ii) separate parameter model in the exponential distribution and (iii) hierarchical Bayes (HB) model via Markov Chain Monte Carlo (MCMC) in the exponential distribution. For the second component, a Polya tree, one of the Bayesian nonparametric distributions, estimates the transition probability. The Polya tree estimation is particularly useful for micro data because it induces smoothing of the micro data or the "thin data" where there is few or no observation for some of the large number of states. This dissertation suggests that a one-dimensional or two-dimensional Polya tree is used, depending on the assumption that the transition probabilities are independent of stock prices. The SCD model is so general that it can simulate changes in volatility or jumps which commonly appear in financial data. The dissertation presents some results on empirical study of the SCD model. First, Monte Carlo simulation generates sample paths for Intel stock prices under this model. Then, the SCD model is applied for option pricing, and the result is compared with prices derived by some existing models such as the Black-Scholes and Binomial tree models. Finally, the dissertation provides the direction of the further research. |
