Exotic Option Pricing in Stochastic Volatility Levy Models and with Fractional Brownian Motion

Abstract

The paper Exotic Option Pricing in Stochastic Volatility Levy Models and with Fractional Brownian Motion aims on extending the restrictive Black-Scholes model by allowing volatility to evolve randomly. These models are used to price exotic derivatives and certificates. The first stochastic volatility model is the Heston model. In order to capture jumps in volatility and stock evolution, Levy processes and Ornstein-Uhlenbeck processes are under discussion. Using the convenient features of Levy processes, a stochastic volatility stock evolution model, where volatility is driven by a Levy process and volatility evolution and stock evolution are linked, is introduced.This model is named after Barndorff-Nielsen and Shephard (BNS for short). However, the BNS model does not include the long memory behavior of volatility. This justifies a discussion about the fractional Brownian motion, the most important stochastic process to model long memory. Finally, the different models are calibrated by fitting prices of observed European options and compared when bonus certificates and express certificates are valued.