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Numerical and BootstrapingTechniques Applied to Option Pricing Formula

Infinite variance distributions are among the competing models used to explain the non-normality of stock price changes (Mandelbrot, 1963; Fama, 1965; Mandelbrot and Taylor, 1967; Rachev and Samorodnitsky, 1993). We investigate the asymptotic option price formula in infinite variance setting for both independent and correlated data using point processes. As we shall see the application of point process models can also leads us to establish a more general option price formula. We also investigate a recursion technique to quantify various characteristics of the resulting formula. This shows that such formulae, and even their approximations, may be difficult to apply in practice. A nonparametric bootstrap method is proposed as one alternative approach and its asymptotic consistency is claimed under a resampling scheme of $m = o(n)$. Some empirical evidence is provided showing the method works in principle, although large sample sizes appear to be needed for accuracy.

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