Option pricing beyond Black–Scholes based on double-fractional diffusion

•We present a new option pricing model with only a few parameters.•We use the generalized approach of anomalous double-fractional diffusion.•We do the analysis on the real data of S&P 500 options.•We compare the double-fractional model with Black–Scholes model and Lévy model.

We show how the prices of options can be determined with the help of double-fractional differential equation in such a way that their inclusion in a portfolio of stocks provides a more reliable hedge against dramatic price drops than the use of options whose prices were fixed by the Black–Scholes formula.