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Laplace Transform Identities for Diffusions, with Applications to Rebates and Barrier Options

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Abstract

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.

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File URL: http://www.business.uts.edu.au/qfrc/research/research_papers/rp203.pdf
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Bibliographic Info

Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 203.

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Length: 23
Date of creation: 01 Oct 2007
Date of revision:
Handle: RePEc:uts:rpaper:203

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Keywords: diffusions; transition densities; first-passage times; Laplce transformations; squared bessel processes; minimal market model; real-world pricing; rebates; barrier options;

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  1. Mark Craddock & David Heath & Eckhard Platen, 1999. "Numerical Inversion of Laplace Transforms: A Survey of Techniques with Applications to Derivative Pricing," Research Paper Series 27, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Antoon Pelsser, 2000. "Pricing double barrier options using Laplace transforms," Finance and Stochastics, Springer, vol. 4(1), pages 95-104.
  3. David Heath & Eckhard Platen, 2002. "Consistent pricing and hedging for a modified constant elasticity of variance model," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 459-467.
  4. Dmitry Davydov & Vadim Linetsky, 2001. "Pricing and Hedging Path-Dependent Options Under the CEV Process," Management Science, INFORMS, vol. 47(7), pages 949-965, July.
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Cited by:
  1. Wong, Bernard, 2009. "Explicit construction of stochastic exponentials with arbitrary expectation k[set membership, variant](0,1)," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 880-883, April.
  2. Mark Craddock & Eckhard Platen, 2009. "On Explicit Probability Laws for Classes of Scalar Diffusions," Research Paper Series 246, Quantitative Finance Research Centre, University of Technology, Sydney.

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