Modulus of continuity of Nemytskii operators with application to the problem of option pricing

Abstract

Abstract We introduce and analyze moduli of continuity for specific classes of Nemytskii operators on spaces of continuous functions, which are given by kernels, strictly monotone in their second argument. Such operators occur as non-linear (outer) mappings for certain problems of option pricing within the Black–Scholes model for time-dependent volatility. This nonlinear mapping can be seen to be continuous, however its convergence properties are poor. Our general results allow to bound the related moduli of continuity, both for the forward and backward non-linear mappings. In particular we explain the observed ill-conditioning of the nonlinear backward problem. The analysis uses some abstract local analysis of index functions, which may be of independent interest.