Convertible bonds are the privileged instrument of corporations willing to rai se capital white benefiting from both debt and equity advantages. They are complex corporate securities which incorporate within a single derivative instrument man y features at the boundary of option theory. Three main research areas emerge: the economic rationale for convertible debt, the hybrid nature of convertible bonds and the analysis of the optimal recall policy lying in the hands of the rational issuer. ln the area of credit risk modeling, we first explore the credit-equity power relationship, an empirical parametrization of credit spreads by stock prices useful to mode! distressed convertible bonds. We provide a theoretical foundation for the credit-equity elasticity involving the financial leverage of the company. We apply the credit-equity power paradigm to derive a general modeling framework for hybrid securities, and we show how to link the variance elasticity to the corporate leverage. Finally, we revisit the valuation of exchangeable bonds by introducing stochastic covariance between the Iwo main sources of risk and introducing the leverage of the underlying entity in the model. ln the area of option pricing, we revisit the modeling of discrete cash dividends which remain notoriously difficult to handle absent an efficient modeling framework. Our approach provides the proper theoretical setting to analyze the dividend protection of convertible bonds, a standard feature since the late 2000s. We apply our modeling approach to the theoretical analysis of the optimal recall policy ofdividend-protected convertible bonds. We show a strong structural incentive for issuing firms to recall and force the conversion of their bonds as soon as possible.