Pricing currency options by generalizations of the mixed fractional brownian motion

Option pricing is an active area in financial industry. The value of option pricing is usually obtained by means of a mathematical option pricing model. Since fractional Brownian motion and mixed fractional Brownian motion processes have some important features in order to get typical tail behavior from financial markets, such as: self-similarity and long-range dependence, they can play a significant role in pricing European option and European currency options. In this thesis, some extensions of the mixed fractional Brownian motion model are proposed to wider classes of pricing options systems. In Chapter 3, a new framework for pricing the European currency option is developed in the case where the spot exchange rate follows a mixed fractional Brownian motion with jumps. An analytic formula for pricing European foreign currency options is proposed using the equivalent martingale measure. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in Chapter 3 as well. Furthermore, the actuarial approach to pricing currency options which transform option pricing into a problem of equivalent of fair insurance premium is introduced. In addition, in Chapter 4, the problem of discrete time option pricing by the mixed fractional Brownian model with transaction costs using a mean self-financing delta hedging argument is considered in a discrete time setting. A European call currency option pricing formula is then obtained. In particular, the minimal pricing of an option under transaction costs is obtained, which shows that time step dt and Hurst exponent H play an important role in option pricing with transaction costs. Finally, Chapter 5 considers the problem of discrete time option pricing by a mixed fractional subdiffusive Black-Scholes model. Under the assumption that the price of the underlying stock follows a time-changed mixed fractional Brownian motion, a pricing formula for the European call option and European call currency option is derived in a discrete time setting with transaction costs.

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