Pricing Double Barrier Options: An Analytical Approach

Double barrier options have become popular instruments in derivative markets. Several papers have already analysed double knock-out call and put options using di erent methods. In a recent paper, Geman and Yor (1996) derive expressions for the Laplace transform of the double barrrier option price. However, they have to resort to numerical inversion of the Laplace transform to obtain option prices. In this paper, we are able to solve, using contour integration, the inverse of the Laplace transforms analytically thereby eliminating the need for numerical inversion routines. To our knowledge, this is one of the rst applications of contour integration to option pricing problems. To illustrate the power of this method, we derive analytical valuation formulas for a much wider variety of double barrier options than has been treated in the literature so far. Many of these variants are nowadays being traded in the markets. Especially, options which pay a xed amount of money (a \rebate") as soon as one of the barriers is hit and double barrier knock-in options.