Option pricing in fractional models

This thesis deals with application of the fractional Black-Scholes and mixed fractional Black-Scholes models to evaluate different type of options. These assessments are considered in four individual papers. In the first articles, the problem of geometric Asian and power options pricing is investigated when the stock price follows a time changed mixed fractional model. In this model, an inverse subordinator process in the mixed fractional Black-Scholes model replaces the physical time. The aim of the third paper is to evaluate the European currency option in a fractional Brownian motion environment by the time-changed strategy. Also, the impact of time step and long range dependence are obtained under transaction costs. Conditional mean hedging under fractional Black-Scholes model is the propose of the second article. The conditional mean hedge of the European vanilla type option with convex or concave positive payoff under transaction costs is obtained. In the fourth article, the mixed fractional Brownian motion with jump process are incorporated to analyze European options in discrete time case. By a mean delta hedging strategy, the pricing model is proposed for European option under transaction costs.