Qualitative financial modelling in fractal dimensions

The Black–Scholes equation is one of the most important partial differential equations governing the value of financial derivatives in financial markets. The Black–Scholes model for pricing stock options has been applied to various payoff structures, and options trading is based on Black and Scholes’ principle of dynamic hedging to estimate and assess option prices over time. However, the Black–Scholes model requires severe constraints, assumptions, and conditions to be applied to real-life financial and economic problems. Several methods and approaches have been developed to approach these conditions, such as fractional Black–Scholes models based on fractional derivatives. These fractional models are expected since the Black–Scholes equation is derived using Ito’s lemma from stochastic calculus, where fractional derivatives play a leading role. Hence, a fractional stochastic model that includes the basic Black–Scholes model as a special case is expected. However, these fractional financial models require computational tools and advanced analytical methods to solve the associated fractional Black–Scholes equations. Nevertheless, it is believed that the fractal nature of economic processes permits to model economical and financial markets problems more accurately compared to the conventional model. The relationship between fractional calculus and fractals is well-known in the literature. This study introduces a generalized Black–Scholes equation in fractal dimensions and discusses its role in financial marketing. In our analysis, we consider power-laws properties for volatility, interest rated, and dividend payout, which emerge in several empirical regularities in quantitative finance and economics. We apply our model to study the problem of pricing barrier option and we estimate the values of fractal dimensions in both time and in space. Our model can be used to obtain the prices of many pay-off models. We observe that fractal dimensions considerably affect the solutions of the Black–Scholes equation and that, for fractal dimensions much smaller than unity, the call option increases significantly. We prove that fractal dimensions are a powerful tool to obtain new results. Further details are analyzed and discussed.