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

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  • Lau, John W.
  • Siu, Tak Kuen

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.

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  • Lau, John W. & Siu, Tak Kuen, 2008. "On option pricing under a completely random measure via a generalized Esscher transform," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 99-107, August.
    • Handle: RePEc:eee:insuma:v:43:y:2008:i:1:p:99-107
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    2. Wenhan Li & Cuixiang Li & Lixia Liu & Mengna Wang, 2021. "Foreign Currency Power Option Pricing Based on Esscher Transform," Computational Economics, Springer;Society for Computational Economics, vol. 58(2), pages 535-548, August.
    3. Tahir Choulli & Ella Elazkany & Mich`ele Vanmaele, 2024. "The second-order Esscher martingale densities for continuous-time market models," Papers 2407.03960, arXiv.org.

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