FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS FOR THE MODELLING OF STOCHASTIC LE´VY PROCESS

Abstract

European-style options for the pricing of financial instruments. The stable fractional L´evy process is used in a new time-fractional L´evy stochastic diffusion equation to reach this goal. The modification involves incorporating an additional term into the L´evy-time fractional diffusion equation, capturing the dynamics of an illiquid market with an impact. We established a new general fractional partial differential equation governing the European option price associated with this new L´evy diffusion equation, which is considered associated with the stable fractional L´evy process. We considered the special density probability function of the L´evy model that modified the equation of the Fourier transform of a European-style option to derive the fractional partial differential equation. Using the obtained new general equation, we present some applications and investigate the numerical analysis of the value of European options. In addition, we analyzed the sensitivity of the option price relative to a number of equation parameters governed by the fractional time L´evy process. The option price is given through a real-valued deterministic function that satisfies some fractional partial differential equations. The weighted-shifted Gr ¨ unwald approximation is the numerical method of the fractional partial differential equation. We used real data to apply our modified model, which was found to be useful and effective in real life. Moreover, we established the fractional Fokker-Planck equation with the L´evy stable process and used the transition probability density function. We modeled market data using stable distribution. We demonstrate the traits and relationships between the fractional Fokker-Planck equation using market data and the simulation approach. The fractional equation within the framework of the fractional stable L´evy equation demonstrated efficacy in pricing European options concerning the risk-free parameter. Using numerical solutions, we examined the dynamics of our fractional equation, which successfully fits realistic market data.