Interactions of early exercise rights and exotic barriers associated with certain derivatives
Creators: name: Ai, Lin
Publisher: name: University of Wollongong
Year: 2025
Subjects: classificationCode: 490106
subject: Financial mathematics
classificationCode: 4901
subject: Applied mathematics

description: Financial derivatives are becoming increasingly popular among investors and academic researchers. As research progresses, more attention is being given to the pricing of complex financial derivatives, such as those incorporating early exercise rights and barrier features.If holders have early exercise rights, they possess the right to exercise their financial derivatives before the maturity date, which is a characteristic feature of American-style derivatives. This early exercise feature transforms the pricing of American-style financial derivatives from a linear to a non-linear problem, unlike the pricing of European-style derivatives, resulting in much greater complexity. Moreover, the existence of early exercise rights contributes to a higher price for American-style financial derivatives compared to their European-style counterparts, due to the additional rights granted to the holders.In terms of exotic barrier features, we explore the “hard” and “soft” barrier features in this thesis. The former refers to the activation of the barrier feature when the recorded time of the underlying asset price over or below a predetermined barrier price exceeds a preset strike time. The latter, also known as the “m out of n” days feature, entails the activation of the barrier feature when the underlying asset price has been above or below a predetermined barrier price for “m” days out of the past “n” consecutive trading days before the current trading day. Both the hard and soft features affect the value of financial derivatives because they introduce the possibility of the derivatives becoming worthless or having a predetermined price.Therefore, if a financial derivative combines both of these two features, namely early exercise rights and exotic barrier features, these two features could interact with each other, influencing the price of the financial derivative. Exploring the interaction between these two features is the goal of our thesis, achieved through pricing a couple of particular exotic derivatives, such as American-style Parasian options and convertible bonds with a soft call.Chapter 1 presents the introduction and literature review on options and convertible bonds, followed by the mathematical background in Chapter 2. Then, we determine the price of a Parasian option with a hard window in Chapter 3 and examine how the early exercise right and barrier feature impact the option price. Subsequently, in Chapter 4, we establish the partial differential equation (PDE) for a Parasian option with a soft window, establishing a price relationship between Parasian options with soft and hard windows. Expanding our focus beyond Parasian options, we also consider convertible bonds, which incorporate both early exercise rights and barrier features. Consequently, we derive the price of “one-touch” callable convertible bonds and explore a property of this financial derivative. Detailed insights into this are provided in Chapter 5. Finally, building on the methodology introduced in Chapter 4, we extend our analysis to determine the value of convertible bonds with a “delayed” barrier feature, which is referred to as a soft call, in Chapter 6. Finally, Chapter 7 contains the conclusions.