Optimal control of perpetual CPDO: minimal cash-out probability and maximal conditional return

In this paper, the optimal leverage function for a perpetual Constant Proportion Debt Obligation (CPDOs) is derived by solving a highly nonlinear Hamilton–Jacobi–Bellman equation with initial conditions. The model to minimize the default probability was built, then the optimal payoff under the condition of minimal default probability is also worked out. The stochastic optimal control theory is utilized to determine the optimal leverage function for the minimization of default probability. Then, the optimal upper bound of the leverage is chosen to obtain a perfect return of the investor. All the equations derived from the above problems have explicit solutions. Based on the solutions, numerical examples are provided, and the relationships between the optimal leverage, as well as the optimal upper bound, and the parameters in the problem, are shown in the figures.