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 1973 and Goldman, Sosin, and 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 and 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‐barrir price with respect to its maturity date. the inversion of the Laplace transform using techniques developed by Geman and Eydeland (1995), is then fairly easy to perform.