This thesis examines the effects of default and market risks on risky or vulnerable content claim instruments including all options, senior and subordinated zero coupon, coupon paying debt securities, and credit derivatives. There are two primary aims of this thesis. The first is to derive valuation equations for these instruments that separates the default or credit risk from the risk free (in terms of default) value by utilising contingent claims analysis and investigating the factors that contribute to default. Default or credit risk is defined as the loss due to counterparty default or the risk that the writer (in the case of options) or issuer (in the case of debt securities) fails to make the promised payment at expiration of the contingent claim. Default risk is dependent on the recovery value of the instrument and the probability of default, both of which are interrelated. The recovery value is assumed to be stochastic in nature. Further refinements of the risks inherent in the stochastic processes are introduced. This involves the inclusion of a stochastic variable, the market portfolio in the stochastic process generating the returns of the recovery value, thus redefining the risks into systematic and specific risk factors. Systematic and specific risk is defined as the adverse movements of assets due to the relationship with the market portfolio and the fluctuations of the market portfolio itself or, due to factors specific to those assets respectively. The second aim of this thesis is to illustrate the uses and applications of the model derived and thus, the main contribution of this thesis,. The approach derived to illustrate these uses and applications is known as Default Premium Approach which focuses not on the current value of the default premium but on its expected value and the confidence levels associated with dispersion of the terminal values. Thus, Default Premium Approach is a form of credit value-at-risk. The Default Premium Approach has two uses. Firstly, as a means for applying equitable capital requirements and secondly, as a quantitative credit risk measurement methodology. These applications of the theoretical model illustrate the strengths of Default Premium Approach in that it has practical consequences and is adaptable to the current requirements of both regulators and market practitioners in their quest to understand and develop equitable capital requirements and a sound risk management methodology that encompasses both default and market risks of example functional instruments.