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