An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $2-\mu$ is time, where $\mu$ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.Comment: 34 pages, 12 figure

Numerical method for pricing governing American options under fractional Black-Scholes model

In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric Lévy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of Crank-Nicolson method and a cubic spline collocation method, respectively. This method exhibits a second order of convergence in space, in time and also has an acceptable speed in comparison with some existing methods. We will compare our results with some recently proposed approaches. Keywords: Geometric Lévy process, fractional Black-Scholes, Crank-Nicolson scheme, Spline collocation, Free Boundary Value Problem

A finite difference method for pricing European and American options under a geometric Lévy process

In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process

Analytically pricing European options under the CGMY model

This thesis investigates the pricing of European-style options under the CGMY model, which can fit the empirically observed data in financial market better than the B-S (Black-Scholes) model. Under this model, the price of options is governed by a FPDE (fractional partial differential equation) with two spatial-fractional derivatives defined in the Weyls sense. In comparison with the derivative of integer order, the fractional-order derivative requires the function value over the entire domain rather than its value at one particular point. This has added an additional degree of diffculty when either the analytical solution or the numerical method is attempted. Albeit difficult, we have managed to derive a closed-form analytical solution for European options under the CGMY model. Based on the solution, we further discuss its asymptotic behaviors and the put-call parity under the adopted CGMY model. Finally, we propose an efficient numerical evaluation technique for the current formula so that it can be easily used in trading practice

Comparison of Numerical Methods on Pricing of European Put Options

Put option is a contract to sell some underlying assets in the future with a certain price. On European put options, selling only can be exercised at maturity date. Behavior of European put options price can be modeled by using the Black-Scholes model which provide an analytical solution. Numerical approximation such as binomial tree, explicit and implicit finite difference methods also can be used to solve Black-Scholes model. Some numerical methods are applied and compared with the analytical solution to determine the best numerical method. The results show that numerical approximations using the binomial tree is more accurate than explicit and implicit finite difference method in pricing European put options. Moreover when the value of T is higher then the error obtained is also higher, while the error obtained is lower when the value of N is higher. The value of T and N cause the increase of the computation time. When the value of T is higher the computation time is lower, while computation time is higher if the value of N is higher. Overall, the lowest computation time is obtained by using an explicit finite difference method with an exceptional as the value of T is big and the value of N is small. The lowest computation time is obtained by using a binomial tree method