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

Three-point compact finite difference scheme on non-uniform meshes for the time-fractional Black-Scholes equation

“…In the next section we construct a compact finite-difference scheme for the TFBSD equation using the fourth-order compact approximation (5) on a three-point stencil of the Tavella-Randal and the quadratic non-uniform meshes and the L1-approximation for the Caputo fractional derivative defined as [10] when U (s, t) is a twice continuously differentiable function [8,10]. From the properties of the Caputo derivative, the TFBSD equation has a natural singularity at t = 0.…”

Laplace Transform Methods for a Free Boundary Problem of Time-Fractional Partial Differential Equation System

“…However, it is argued that the time derivative / should be replaced by the fractional derivative / (0 < ≤ 1) under the assumption that the change in the option price follows a fractal transmission system (see, e.g., [32][33][34]). So, the time-fractional American put options satisfy the following system of PDEs:…”

“…In this paper, we study the pricing of American options with time-fractional model which has essential difference to the space-fractional model. Following the model in [33], we assume that the underlying asset price still follows the classical Brownian motion, but the change in the option price is considered as a fractal transmission system. The price of such American 2 Discrete Dynamics in Nature and Society option follows time-fractional partial differential equations (PDEs) with free boundaries.…”

“…The price of such American 2 Discrete Dynamics in Nature and Society option follows time-fractional partial differential equations (PDEs) with free boundaries. The solution of the timefractional PDEs with two free boundaries is more challenging than solving the fractional PDEs with fixed boundary for European option pricing in [33] and much more complex than solving single Black-Scholes-Merton differential equation in [32].…”

An adaptive moving mesh method for a time-fractional Black–Scholes equation

“…There are a few analytical and numerical methods for the valuation of the time-fractional B-S equations. The analytical methods for the time-fractional B-S equations are usually based on integral transform methods [5,12,26], homotopy analysis methods [17], or wavelet based hybrid methods [11], etc. But the solutions obtained by analytical methods usually take the form of a convolution of some special functions or an infinite series with an integral, which makes them hard to calculate.…”