Super Replication and Uncertain Volatility: European, American and Passport Options

Abstract

Despite a large number of works, there is no unique way to model the stochastic volatility of an underlying. A new approach has been developed where one determines the (super)hedging strategies which are admissible for any model where the volatility is lying in a known interval without other restriction.Our first objective standardizes and to develops the existing results on European options in a common setting. A second objective is the study of American options. We characterize the price by a stochastic control problem with a control over volatility and over stopping times. A last objective is the study of European and American passport options. The prices are characterized by the same kind of stochastic control problem with moreover a control over all the trading strategies of the option's buyer.All those characterizations are valid for only very smooth payoff functions. We extend them to the practical case where the payoff functions are merely continuous. With an analytic point of view, the computation of those prices is given by the resolution of Hamilton-Jacobi-Bellman (HJB) equations (European case) or of a system of HJB equations (American case) with an initial condition.