Valuation of Continuously Monitored Double Barrier Options and Related Securities

Abstract

In this article we apply Carr's randomization approximation and the operator form of the Wiener-Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain contingent claims with first-touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options.Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Levy-driven models including Variance Gamma processes, Normal Inverse Gaussian processes and KoBoL processes (a.k.a. the CGMY model). At the same time, our work gives new insight into the known explicit formulas obtained by other authors in the setting of the Black-Scholes model. The operator form of the Wiener-Hopf method is generalized for wide classes of processes including the important class of Variance Gamma processes.Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock-out double barrier put/call options as well as double-no-touch options.