TY - JOUR

T1 - On option pricing under a completely random measure via a generalized Esscher transform

AU - Lau, John

AU - Siu, T.K.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

U2 - 10.1016/j.insmatheco.2008.03.006

DO - 10.1016/j.insmatheco.2008.03.006

M3 - Article

VL - 43

SP - 99

EP - 107

JO - Insurance Mathematics & Economics

JF - Insurance Mathematics & Economics

SN - 0167-6687

IS - 1

ER -