A Black-Scholes inequality: applications and generalisations (Q2282961)

The paper deals with the far generalization of the classical Black-Scholes model of option pricing. The main objective are so called ``call price surfaces'' ie. functions \[ C:[0,\infty)^2\longrightarrow [0,1] \] given by a formula \[ C(\kappa, t)= E((\alpha_t - \kappa \beta_t)^+), \] where \((\alpha_t,\beta_t)_{t\geq 0}\) is a pair of nonnegative martingales such that \(\alpha_0=1=\beta_0\). The author provides several characterizations of such objects, methods of construction and study their properties. Special attention is paid to a semigroup structure. The practical interest of the presented results is driven by the fact that \(C\) is modelling a price of an option divided by the initial value of the base asset, hence an observable object (at least for some \(t\) and \(\kappa\)).