Theory and Estimation of Drift Dynamics of Asset Price Process

Abstract

This project aims at investigating the behaviour of drift dynamics in financial models. Earlier models used geometric Brownian motion (GBM) for capturing the behaviour of stock prices. In 1973, Black, Scholes and Merton used the GBM model for pricing European style options and also gave a mathematical formula for the same. The flaw of this model was it took several market parameters as volatility, drift, interest rate as a constant which need not be the case with the actual market. Although this model was a flawed one, it provided the much-needed base required in modelling the market and motivated further research. To rectify these shortcomings, after a significant amount of research in the field of mathematical finance, several models were proposed and one such, the Markov modulated GBM model which takes the market parameters as a function of a Markov chain which evolves according to a specific transition rate. This opens a wide range of possibilities and ideas for research. This project deals with surveying several of the key process involved in these models and simulating several of them. Also, several numerical experiments related to the theoretical results were carried out and analyzed. The problem of estimating the drift parameter in these models was investigated using theoretical calculations and experimenting them numerically.