NIG-Lévy approximations and copula based bivariate option pricing

Abstract

We set out to find how copula-based dependence modelling of large jumps in the NIG-Lévy model affects the option price. This leads to the definition of a rejection-based approximation algorithm for the NIG-Lévy. Using small jumps approximation provided by theory from Asmussen and Rosinki, an automatic algorithm is developed. Furthermore, the empirical copula of two dependent processes is extracted, and the effect of the copula on large jumps is analysed for a given approximation level. In general, it can be observed that the small jumps dominate the dependence structure, as well as the behavior of the process. An observation with regards to thin tails is made: The dependent jumps produced by the conditional copula distribution vary a lot even in the 4th or 5th significant digit. Therefore, one can see indications that for certain parameter sets, copulas with sufficient spread in the tails may not be ideal for modelling the dependence in the normalized Lévy measure. Finally, a discussion on the effect of the large jump copula on option prices is presented, and compared to the option price of independent processes. An approach to finding the Esscher parameters is presented, and the issues using an Esscher transform is presented. The findings indicate that copulas on large jumps can have a bigger effect on the option prices if dependency is low, as variation in large jumps can be bigger for the dependent process than the original process. Option prices in this case is expected to be closer to the prices found of the basket of independent assets. For high dependency, the option prices are generally lower that that of the independent case, as dependence restrict the movement of the second process.