General properties of the Solutions to Moving Boundary Problems for Black-Sholes Equations

We study general properties such as the solution representation of a moving boundary value problem of the Black-Scholes equation, its min-max estimation, lower and upper gradient estimates, and strict monotonicity with respect to the spatial variables of the solution. These results are used in the study of a structural model of pricing puttable bond with credit risk. We first prove the solution representation of a special fixed boundary value problem of the Black-Scholes equation, the min-max estimate, the lower and upper gradient estimates, and the strict monotonicity with respect to the spatial variables of the solution. Then, these results are applied to give the solution representation of a moving boundary value problem of the Black-Scholes equation with moving boundary in the form of an exponential function, the min-max estimate, the lower and upper gradient estimates, and the strict monotonicity results on the spatial variables of the solution. Finally, we illustrate how these results can be used in the derivation of analytical pricing formulae and financial analysis of price functions of puttable bonds with credit risk (corporate bonds with one early redemption date). Our results can be used for the derivation and analysis of the analytical pricing formulae of the one-factor structural model of a more general puttable bonds with credit risk (corporate bond with several early redemption dates).