Pricing double barrier options using Laplace transforms

In this paper we address the pricing of double barrier options. To derive the density function of the first-hit times of the barriers, we analytically invert the Laplace transform by contour integration. With these barrier densities, we derive pricing formulæ for new types of barrier options: knock-out barrier options which pay a rebate when either one of the barriers is hit. Furthermore we discuss more complicated types of barrier options like double knock-in options.