Stochastic volatility enhanced lévy processes in financial asset prcing. Pricing European call options

Abstract

This report investigates several stochastic processes used for pricing European call options. The pure jump L´evy processes are the cornerstone in the different models, here presented. These do not have a Brownian motion component, therefore the stochastic volatility is instead introduced as a stochastic time-changing effect. In the paper“Stochastic volatility for L´evy processes”written by Carr, Geman, Madan and Yor, the types of stochastic time-changed mean corrected exponential L´evy processes (type 2 models) used are claimed to be martingales without proof. In the book “Financial modelling with jump processes”written by Cont and Tankov, an attempt to prove the martingale property of these has been given but is insufficient. In this report, a proof of the martingale property is made and presented. Additionally, mean corrected stochastically time-changed exponential L´evy processes (type 1 models) are introduced as proposed by Carr, Geman, Madan and Yor. The models are calibrated against OMXS30 European call options and the calibration performances of the different models are evaluated.