Numerical pricing of American options on extrema with continuous sampling

Abstract

One of the typical option classes is formed by lookback options whose values depend also on the extrema of the underlying asset over a certain period of time. Moreover, incorporating the American constraint, which admits early exercise, has increased the popularity of these hedging and speculation instruments over recent years. In this paper, we consider the problem of pricing continuously observed American-style lookback options with fixed strike. Since no analytic formulae exist for this case, we follow an approach that formulates the corresponding option pricing problem as the parabolic partial differential inequality subject to a constraint, handled by a penalty technique. As a result, we obtain the pricing equation restricted to a triangular domain, where the path-dependent variable appears as a parameter only in the initial and boundary conditions. The contribution of the paper lies in the proposal of a numerical scheme that solves this option pricing problem. The numerical technique proposed arises from the dis- continuous Galerkin that enables easy implementation of penalties and weak enforcement of boundary conditions. Finally, the capabilities of the numerical scheme are demonstrated within a simple empirical study on the reference experiments.