Fractal barrier option pricing under sub-mixed fractional Brownian motion with jump processes

In this work, we mainly focused on the pricing formula for fractal barrier options where the underlying asset followed the sub-mixed fractional Brownian motion with jump, including the down-and-out call option, the down-and-out put option, the down-and-in call option, the down-and-in put option, and so on. To start, the fractal Black-Scholes type partial differential equation was established by using the fractal Itô's formula and a self-financing strategy. Then, by transforming the partial differential equation to the Cauchy problem, we obtained the explicit pricing formulae for fractal barrier options. Finally, the effects of barrier price, fractal dimension, Hurst index, jump intensity, and volatility on the value of fractal barrier options were exhibited through numerical experiments.