Document Type

Thesis

Degree Name

Master of Science (MSc)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Joe Campolieti

Advisor Role

Thesis Supervisor

Abstract

In this thesis we focus on the development of a new class of stochastic models for asset price processes and their application to option pricing and hedging. The asset price process involves analytical treatments for calculating first-hitting (or first-passage) times for a regular diffusion with killing in combination with Markov state-switching. The dynamics is naturally dictated by the underlying diffusion process itself rather than arising from some addition exogenous process. To date, this class of asset pricing models appears to be novel in the literature and, moreover, offers a significant to the standard geometric Brownian motion commonly used in the original Black-Scholes (BS) model. In the class of mdoels to be developed and analyzed in this thesis, model-specific first-hitting times are introduced (e.g. for upper and lower barriers) for underlying asset price diffusion in continuous time with an additionally embedded two-state (extendible to multiple-state) continuous-time Markov chain. In each state, the asset price process obeys geometric Brownian motion with a constant volatility corresponding to the variation of the process. Upper and lower state switching barriers are then introduced as natural barriers separating these regions. The random (stopping) times for the process to switch from one state to another are its first-hitting times (i.e. barrier crossing times from below and from above). We then investigate the analytical tractability for computing transition probability densities, first-passage time densities and for computing European option pricing formulae. Model calibration analysis is also studied.

Convocation Year

2008

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