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Abstract

The objective of this thesis is to improve the understanding of the models arising in convertible bond (CB) valuation, introduce new models incorporating interest rate and credit risk and develop sophisticated numerical methods to implement those models. We carry out our analysis in the CB market because it is rapidly increasing and yet not enough research has been done to accurately and efficiently price those instruments. Moreover, the complexity of the mathematical models arising in CB valuation make this area of finance a particularly challenging and interesting one to research. Despite concentrating on CB pricing we believe that our work has broader implications. This is because we proposed a very general and flexible framework that could be applied to price any American-style contingent claim in a two-factor setting.
In the first part of the thesis we introduce for the first time in finance the method of characteristics/finite elements combined with a Lagrange multiplier method to solve two-factor pricing models for financial derivatives. To demonstrate the applicability of the approach, we solve a convertible bond model with equity and interest rate risk; we focus on the consistent and rigorous specification of the model, and fully address its practical implementation.

The second part of the thesis explores how to incorporate credit risk in CB valuation. This is a complex task due to the hybrid nature of CBs and there is no consensus in the literature of whether it has to be done in an equity or an asset based framework. For this reason, we introduce new equity based and asset based models, which include stochastic interest rate, and solve them using the numerical technique developed in the first part. Regarding the equity based approach we propose a unified intensity-based framework of which most existing comparable models are special cases; this allows us to implement and analyze previous models as well as introduce new ones. We find that different models lead to significantly different prices and that it is important to consistently specify the process for the stock price, the recovery value and the holder’s rights upon default. The flexibility of the approach enables us to generate a great number of default-recovery scenarios. Regarding the asset based approach, we introduce a new model which has both structural and reduced form features and in which recovery is endogenised. Despite the fact that the state variable is unobserved and a simple capital structure is assumed, the possibility that default can be triggered both exogenously and endogenously at a cash-flow time leads to a more realistic formulation than it can be achieved in an equity based approach. Moreover our endogenised recovery potentially allows a greater ability to estimate recovery values from market data.

We conclude that both models have a great potential to explain empirical CB values and given their advantages and disadvantages can be chosen depending on the purpose and the circumstances.

Publication Type:	Thesis (Doctoral)
Subjects:	H Social Sciences > HJ Public Finance
Departments:	Bayes Business School > Bayes Business School Doctoral Theses Bayes Business School > Finance Doctoral Theses

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