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Valuing Continuous Barrier Options on a Lattice solution for a Stochastic Dirichlet Problem ¤

By Grace Kuan and Nick Webber

Abstract

The stochastic Dirichlet problem computes values within a domain of certain functions with known values at the boundary of the domain. When applied to valuing barrier options, solutions are expressed as expected discounted payo¤s achieved at hitting times to the boundary of the domain. We construct a lattice solution to the stochastic Dirichlet problem. In between time steps on the lattice, the lattice process is assumed to have the bridge distribution of the underlying stochastic process. We apply the Dirichlet lattice to valuing barrier options. A plain simple scheme converges very slowly. We …nd that the Dirichlet lattice is considerably faster than a plain lattice scheme, converging to 2 decimal places in only several hundred time steps. The Dirichlet lattice can directly value knock-in barrier options, including knock-in Bermudan barrier options which cannot normally be priced by a plain lattice method. It values Bermudan barrier options and barrier options with non-linear barriers equally quickly. We present results demonstrating the superiority of the Dirichlet lattice over both a plain lattice method and a conditional Monte Carlo method. We wish to thank participants at a City University seminar for comments on an early version of this paper. The paper has also bene…ted from conversations with Gordon Gemmill and Paul Dawson. 1

Year: 2003
OAI identifier: oai:CiteSeerX.psu:10.1.1.199.4532
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