Pricing and Hedging Double - Barrier Options : A Probabilistic Approach '

Barrier options have become increasingly popular over the last few years. Less expensive than standard options, they may provide the appropriate hedge in a number of risk management strategies. In the case of a single barrier option, the valuation problem is not very difficult (see Merton 193, Goldman-Sosin-Gatto 1979). The situation where the option gets knocked out when the underlying instrument hits either of two well-defined boundaries is less straightforward. Kunitomo and Ikeda (1992) provide a pricing formula expressed as the sum of an infinite series whose convergence is studied through numerical procedures and suggested to be rapid. We follow a methodology which proved quite successful in the case of Asian options (see Geman-Yor 1992, 1993) and which has its roots in some fundamental properties of Brownian motion. This methodology permits the derivation of a simple expression of the Laplace transform of the double-barrier price with respect to its maturity date. The inversion of the Laplace transform is then fairly easy to perform, using the techniques developed by Geman-Eydeland ( 1995).