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

Hardy 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 -