Pricing Double Barrier Options: An Analytical Approach

Double barrier options have become popular instruments in derivative markets. Several papers_new have already analyseddouble knock-out call and put options using different methods. In a recent paper, Geman and Yor (1996) deriveexpressions for the Laplace transform of the double barrrier option price. However, they have to resort to numericalinversion 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 ourknowledge, this is one of the first applications of contour integration to option pricing problems. To illustrate the power ofthis method, we derive analytical valuation formulas for a much wider variety of double barrier options than has beentreated in the literature so far. Many of these variants are nowadays being traded in the markets. Especially, options whichpay a fixed amount of money (a "rebate") as soon as one of the barriers is hit and double barrier knock-in options.