Sub mixed fractional Brownian motion and its application to finance

The paper focuses on the valuation of the European contract as a vanilla option and the Arithmetic Asian contract as an exotic option, where the underlying asset price is driven by the sub mixed fractional Geometric Brownian motion model with multiple sources of risk. The study begins by conducting a comprehensive analysis of market price data to extract relevant properties using statistical tests. Subsequently, we present the sub-mixed fractional Brownian motion model with multiple sources of risk and demonstrate its superiority in predicting market behavior compared to standard and mixed fractional Brownian motion models according to the Tesla and the Meta Platforms companies stock prices data. Then, we use Ito's formula for sub-mixed fractional Geometric Brownian motion and the Fokker–Planck equation to derive a closed-form solution for the European call option price. After that, we compare the option price under different models based on the Tesla and the Meta Platforms companies European call option prices. Moreover, we approximate the Arithmetic Asian option price under the presented model. To do so first, we obtain a closed form solution for the Geometric Asian call option. Then, since the option lacks a closed-form solution, we utilize the Control variate Monte Carlo simulation method with the geometric Asian option price serving as a control variate variable to estimate the Arithmetic Asian option price. Finally, we compare the option price under different models.