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## Laplace transform identities for diffusions, with applications to rebates and barrier options

### Volume 83 / 2008

Banach Center Publications 83 (2008), 139-157 MSC: Primary 60J60, 91B28; Secondary 44A10, 47D07, 60J70. DOI: 10.4064/bc83-0-9

#### Abstract

We start with a general time-homogeneous scalar diffusion whose state space is an interval $I\subseteq\mathbb R$. If it is started at $x\in I$, then we consider the problem of imposing upper and/or lower boundary conditions at two points $a,b\in I$, 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.

#### Authors

• Hardy HulleySchool of Finance and Economics
University of Technology, Sydney
P.O. Box 123, Broadway, NSW 2007, Australia
e-mail
• Eckhard PlatenSchool of Finance and Economics and Department of Mathematical Sciences
University of Technology, Sydney
P.O. Box 123, Broadway, NSW 2007, Australia
e-mail

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