Abstracts

The quest for an analytical solution to complex derivatives such as double-barrier options has always been a challenge in the literature. Geman and Yor (1996) derive expressions for the Laplace transform of the double barrrier option price. However, they have to use a numerical inversion of the Laplace transform to obtain an approximate solution to option prices, thereby involving a tedious calculation. We study the price of complex derivatives, namely double barrier options proposing a new methodology that represents the solution of heat equations in the form of Volterra integral equations of the second kind. We provide an approximate solution whose error has an exponential decay allowing very rapid convergence with a few iterations only. Given the power of our approach we think that many more applications will follow, whilst Geman and Yor (1996) cover one type of barrier options at a time only: knock-out calls and puts. We provide some numerical applications and compare our results to German-Yor and to Kunitomo and Ikeda (2000) for double barrier knock-out options with no rebate and to the Crank-Nicholson finite difference and Kunitomo-Ikeda method for double barrier options that do pay a rebate at hit.