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We consider the asymptotic option pricing formula under an infinite variance paradigm using a randomized version of the Cox-Ross-Rubinstein binomial option pricing approach. We discuss practical difficulties in applying the asymptotic formula and suggest a non-parametric bootstrap as an estimation technique. Using point process theory, the asymptotic consistency of the bootstrap approach is established under a resampling scheme of m = o(n). We briefly discuss extensions to correlated data and show the option pricing formula may no longer be valid in such settings. Finally, we consider a finite variance setting involving innovations from a variance gamma process. We derive the asymptotic option pricing formula and show that the non-parametric bootstrap is consistent.

In the present paper, starting with the Black-Scholes equations, whose solutions are the values of European options, we describe the exponential jump-diffusion model of Levy process type. Here, a jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general "partial integro-differential equation" (PIDE). With a combined compact difference (CCD) scheme for the spatial discretization, a high-order method is proposed for solving exponential jump-diffusion models. The method is sixth-order accurate in space and second-order accurate in time. A known analytical solution to the model is used to evaluate the performance of the numerical scheme.

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