MPS-RR 2000-46
December 2000

Barrier Options and Touch-and-out Options under Regular Lévy Processes of Exponential Type

Svetlana Boyarchenko, Sergei Levendorskii

Abstract

We derive explicit formulas for barrier options of European type and touch-and-out options assuming that under a chosen equivalent martingale measure the stock returns follow a Lévy process from a wide class, which contains Brownian Motions (BM), Normal Inverse Gaussian Processes (NIG), Hyperbolic Processes (HP) and Truncated Lévy Processes (TLP), and any finite mixture of independent BM, NIG, HP and TLP. In contrast to the Gaussian case, for a barrier option, a rebate must be specified not only at a barrier but for all values of the stock the other side of the barrier, the reason being that trajectories of a non- Gaussian Lévy process are discontinuous. We consider options with the constant or exponentially decaying rebate, and options which pay a fixed rebate when the first barrier has been crossed but the second one has not. We obtain pricing formulas by solving corresponding boundary problems for the generalized Black-Scholes equation. We use the connection between the resolvent and the infinitesimal generator of the process, the representation theorem for analytic semigroups, the Wiener-Hopf factorization method and the theory of pseudo-differential operators.

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