On option pricing under a completely random measure via a generalized Esscher transform

John W. Lau, Tak Kuen Siu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771-1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416-440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.

Original languageEnglish
Pages (from-to)99-107
Number of pages9
JournalInsurance: Mathematics and Economics
Volume43
Issue number1
DOIs
Publication statusPublished - Aug 2008
Externally publishedYes

Keywords

  • Esscher transform
  • Generalized Gamma processes
  • Kernel-biased completely random measures
  • Laplace functionals
  • Option pricing

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