MPS-RR 2000-46

December 2000

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

by:

### Svetlana Boyarchenko, Sergei Levendorskii

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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This paper has now been published in *Ann. Appl. Probab. 12, 1261--1298 (2002)*