Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment

The Application of Fractional Brownian Motion in Option Pricing

In this text, Fractional Brown Motion theory during random process is applied to research the option pricing problem. Firstly, Fractional Brown Motion theory and actuarial pricing method of option are utilized to derive Black-Scholes formula under Fractional Brown Motion and form corresponding mathematical model to describe option pricing. Secondly, based on BYD stock, estimation model on volat...

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

On the Mixed Fractional Brownian Motion

If H = 1/2, BH is the ordinary Brownian motion denoted by B = {Bt, t ≥ 0}. Among the properties of this process, we recall the following: (i) B 0 = 0P-almost surely; (ii) for all t ≥ 0, E((B t )2)= t2H ; (iii) the increments of BH are stationary and self-similar with order H ; (iv) the trajectories of BH are almost surely continuous and not differentiable (see [7]). Let us take a and b as two r...

Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland's hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, th...

Pricing Model for Convertible Bonds: A Mixed Fractional Brownian Motion with Jumps

A mathematical model to price convertible bonds involving mixed fractional Brownian motion with jumps is presented. We obtain a general pricing formula using the risk neutral pricing principle and quasi-conditional expectation. The sensitivity of the price to changing various parameters is discussed. Theoretical prices from our jump mixed fractional Brownian motion model are compared with the p...