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This paper considers pricing European options in a large class of one-dimensional Markovian jump processes known as subordinate diffusions, which are obtained by time changing a diffusion process with an independent Lévy or additive random clock. These jump processes are non-Lévy in general, and they can be viewed as natural generalization of many popular Lévy processes used in finance. Subordinate diffusions offer richer jump behavior than Lévy processes and they have found a variety of applications in financial modelling. The pricing problem for these processes presents unique challenges as existing numerical PIDE schemes fail to be efficient and the applicability of transform methods to many subordinate diffusions is unclear. We develop a novel method based on finite difference approximation of spatial derivatives and matrix eigendecomposition, and it can deal with diffusions that exhibit various types of boundary behavior. Since financial payoffs are typically not smooth, we apply a smoothing technique and use extrapolation to speed up convergence. We provide convergence and error analysis and perform various numerical experiments to show the proposed method is fast and accurate. Extension to pricing path-dependent options will be investigated in a follow-up paper.

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Mijatović and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.

We experimentally investigated dissociative double ionization of CO 2 induced by intense femtosecond laser pulses. Three-dimensional momenta were precisely measured for correlated fragmental ions CO + and O + . The dissociation dynamics of CO 2 2+ was theoretically simulated by using a Coulomb potential approximation. The ultrashort dissociation time (∼400 fs) can well explain the observed anisotropic angular distribution of fragmental ions relative to the laser polarization.