Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Giuseppe Campolieti

Advisor Role

Supervisor

Abstract

In this dissertation, we investigate the properties of last passage times and excursion theory in one-dimensional solvable diffusions, emphasizing their applications in financial modeling, particularly in option pricing. We derive closed-form formulas for the marginal distribution of last passage times and their joint distribution with process values, including the maximum and minimum of the process value. The focus is on time-homogeneous diffusions with various boundaries and imposed killing. Employing spectral expansion theory, we derive explicit formulas for distributions of last passage times in common processes such as Drifted Brownian Motion (BM), Squared Bessel (SQB), Ornstein-Uhlenbeck (OU), and Cox-Ingersoll-Ross (CIR) models. By simple mapping techniques, we also derive distributions for Geometric Brownian Motion (GBM) and Constant Elasticity of Variance (CEV) processes. Additionally, by applying a diffusion canonical transformation, we extend our study to multi-parameter families of solvable diffusions with nonlinear local volatility, including the so-called Bessel-K and Unbounded OU (UOU) processes. Our findings are substantiated with numerical calculations to verify accuracy.

The practical application of these theoretical insights is explored in the realm of option pricing. We focus on barrier options related to the last passage time, particularly introducing and analyzing new types of step options. Utilizing our established joint distribution formulas, we compute the initial prices of these options under common asset pricing models, supported by extensive numerical calculations.

An additional novel contribution of this dissertation lies in the exploration of excursion theory. We examine the relationship between last passage time and the first hitting time after a fixed time $T$, deriving general formulas for all solvable diffusions. Our comprehensive analysis extends to related processes, like the excursion process straddling a time $T$ and the meander processes, for which we derive formulas for their joint distribution with the last passage time. We apply these formulas to the Brownian Motion case as a validation, successfully recovering known results.

Convocation Year

2024

Convocation Season

Spring

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Available for download on Wednesday, January 29, 2025

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