  The Constant Elasticity of Variance (CEV) model is mathematically presented
and then used in a Credit-Equity hybrid framework. Next, we propose extensions
to the CEV model with default: firstly by adding a stochastic volatility
diffusion uncorrelated from the stock price process, then by more generally
time changing Bessel processes and finally by correlating stochastic volatility
moves to the stock ones. Properties about strict local and true martingales in
this study are discussed. Analytical formulas are provided and Fourier and
Laplace transform techniques can then be used to compute option prices and
probabilities of default.
