Analytically pricing double barrier options based on a time-fractional Black–Scholes equation

Abstract

This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this scenario, the option price is governed by a modified Black–Scholes equation with a time-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. The existence of the time-fractional derivative, in conjunction with the presence of two barriers of the double-barrier options, has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Albeit difficult, we have managed to find an explicit closed-form analytical solution for double-barrier options, which has been taken to price the single barrier options and European path-independent options under the same framework as a special case of the current solution. In addition, not only have we provided a theoretical proof for the convergence of the newly-found analytic solution in series form, which is a vital step to show the closeness of our solution, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice.