General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

摘要

We provide representations of solutions to terminal value problems ofinhomogeneous Black-Scholes equations and studied such general properties asmin-max estimates, gradient estimates, monotonicity and convexity of thesolutions with respect to the stock price variable, which are important forfinancial security pricing. In particular, we focus on finding representationof the gradient (with respect to the stock price variable) of solutions to theterminal value problems with discontinuous terminal payoffs or inhomogeneousterms. Such terminal value problems are often encountered in pricing problemsof compound-like options such as Bermudan options or defaultable bonds withdiscrete default barrier, default intensity and endogenous default recovery.Our results are applied in pricing defaultable discrete coupon bonds.