  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).
