ADVANCES IN MATHEMATICS OF FINANCE BANACH CENTER PUBLICATIONS, VOLUME 83 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2008 LAPLACE TRANSFORM IDENTITIES FOR DIFFUSIONS, WITH APPLICATIONS TO REBATES AND BARRIER OPTIONS HARDY HULLEY School of Finance and Economics University of Technology, Sydney P.O. Box 123, Broadway, NSW 2007, Australia E-mail: hardy.hulley@uts.edu.au ECKHARD PLATEN School of Finance and Economics and Department of Mathematical Sciences University of Technology, Sydney P.O. Box 123, Broadway, NSW 2007, Australia E-mail: eckhard.platen@uts.edu.au Abstract. We start with a general time-homogeneous scalar diffusion whose state space is an interval I ⊆ R. If it is started at x ∈ I , then we consider the problem of imposing upper and/or lower boundary conditions at two points a, b ∈ I , where a<x<b. Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model. 1. Introduction. The theory of time-homogeneous linear scalar diffusions is elegant and classical, with Borodin and Salminen [4], Itˆ o and McKean [16] and Karlin and Taylor [17] 2000 Mathematics Subject Classification : Primary 60J60, 91B28; Secondary 44A10, 47D07, 60J70. Key words and phrases : diffusions, transition densities, first-passage times, Laplace trans- forms, squared Bessel processes, minimal market model, real-world pricing, rebates, barrier options. The paper is in final form and no version of it will be published elsewhere. [139] c  Instytut Matematyczny PAN, 2008