Under discrete-time GARCH models markets are incomplete so there is more than one price kernel for valuing contingent claims. This motivates the quest for selecting an appropriate price kernel. Different methods have been proposed for the choice of a price kernel. Some of them can be justified by economic equilibrium arguments. This paper studies risk-neutral dynamics of various classes of Generalized Hyperbolic GARCH models arising from different price kernels. We discuss the properties of these dynamics and show that for some special cases, some pricing kernels considered here lead to similar risk neutral GARCH dynamics. Real data examples for pricing European options on the S&P 500 index emphasize the importance of the choice of a price kernel.

The objective of this work is to present an ‘arbitrage statistics’ strategy that trades Call options in Brazilian derivatives market. The developed algorithm performs an valuation of the nominal values of a random variable (Z), based on Radon-Nikodým derivative, together with a pre-selection of trades forming a subset of the database base selected from a parametric analysis over the model. To build the algorithm, a database of PETR4 Call options traded from January 2th, 2015 to December 11 th, 2019 was used, as well as all closing prices of the PETR4 share for the same period. From the data consolidated we realize a fitting process on the distribution of returns of PETR4 for a mixture of gaussian densities distribution (here called Mistura de Normais distribution), considering the Kolmogorov-Smirnov test, obtaining this way an empirical observed probability, in other words, the probability measure of the “real world” (P). In parallel, from the pricing formula of a Call option we develop algebraically a way to estimate the probability of occurrence that an option be superior to its strike, in other words, the probability measure in the risk-neutral world ( Q ). Based on these values, the aleatory variable Z (ratio between the probability measures Q  and P) is calculated and the strategy is applied by dividing in two parts: first, the parametric assessment of the each factor analysis and the execution of a sensitivity analysis over the results. Second, applying the “refined” strategy (adjusted based on the previous assessment) considering the combination between each different factor from the previous phase, obtaining the final results. The quality assessment of the strategy is based on the distribution of returns and the calculation of the median and average of the returns from the simulations. Another objective of this study is to suggest to financial market an arbitrage statistics strategy for derivatives market that is independent of the choice of the option pricing model (“model free”) and that is able to explore possible discrepancies and singularities of the Brazilian stock and options market.