Pricing of Lookback Options with NIG and VG Dynamics, using the Numerical Path Integration Method.

Abstract

In this thesis we explore the use of the Normal Inverse Gaussian (NIG) and the Variance Gamma (VG) distribution to model stock returns. We compare the NIG market model and the VG market model with historical financial data, and we calibrate both models to European Vanilla call options observed in the market. We show how discretely monitored fixed strike lookback call options and up-and-out barrier call options can be calculated fast and accurately under both the NIG market model and the VG market model, using the numerical path integration method. We introduce the numerical path integration as a convolution integral and calculate the integral by the fast Fourier transform (FFT), leading to a pure multiplication in the Fourier domain. Some basic definitions, theorems and results of the Fourier transformation theory are reviewed.