General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs

We provide representations of solutions to terminal value problems of inhomogeneous Black–Scholes equations and study such general properties as min–max estimates, gradient estimates, monotonicity and convexity of the solutions with respect to the stock price variable, which are important for financial security pricing. In particular, we focus on finding representation of the gradient (with respect to the stock price variable) of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms. Such terminal value problems are often encountered in pricing problems of compound-like options such as Bermudan options or defaultable bonds with discrete default barrier, default intensity and endogenous default recovery. Our results can be used in pricing real defaultable bonds under consideration of existence of discrete coupons or taxes on coupons.