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

Banach Center Publications (2008)

- Volume: 83, Issue: 1, page 139-157
- ISSN: 0137-6934

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topHardy Hulley, and Eckhard Platen. "Laplace transform identities for diffusions, with applications to rebates and barrier options." Banach Center Publications 83.1 (2008): 139-157. <http://eudml.org/doc/281784>.

@article{HardyHulley2008,

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.},

author = {Hardy Hulley, Eckhard Platen},

journal = {Banach Center Publications},

keywords = {diffusions; transition densities; first-passage times; Laplace transforms; squared Bessel processes; minimal market model; real-world pricing; rebates; barrier options},

language = {eng},

number = {1},

pages = {139-157},

title = {Laplace transform identities for diffusions, with applications to rebates and barrier options},

url = {http://eudml.org/doc/281784},

volume = {83},

year = {2008},

}

TY - JOUR

AU - Hardy Hulley

AU - Eckhard Platen

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

JO - Banach Center Publications

PY - 2008

VL - 83

IS - 1

SP - 139

EP - 157

AB - 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.

LA - eng

KW - diffusions; transition densities; first-passage times; Laplace transforms; squared Bessel processes; minimal market model; real-world pricing; rebates; barrier options

UR - http://eudml.org/doc/281784

ER -