Risk Neutral Jump Arrival Rates Implied in Option Prices and Their Models

Characteristic functions of risk neutral densities are constructed from the prices of options at a fixed maturity using well-known procedures. The logarithm of these characteristic functions are shown to synthesize the Fourier transform of jump arrival tails. The formal arrival rate tails are actual arrival rates if their derivatives have an appropriate sign. The derivatives of formal arrival rate tails embedded in option prices are observed on occasion to be negative, reflecting signed jump arrival rates. Although puzzling at first, we further observe that simple analytical cosine perturbations of the symmetric variance gamma Lévy density provides theoretical examples of such signed arrival rates consistent with a probability density. Additionally signed arrival rates also arise when models of signals perturbed by independent noise yield examples of characteristic functions for signal densities that are ratios of pure jump infinitely divisible characteristic functions. Such ratio characteristic functions can reflect signed arrival rates. Specific models using ratios of bilateral gamma and CGMY models are developed and calibrated to short maturity option prices. The ratio models provide significant improvements over their non-ratio counterparts. The models fall in the class of what have recently been termed to be quasi-infinitely divisible distributions.