Abstract
A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution.
Highlights
The valuation of options based on stochastic processes dates back to 1877, when Charles Castelli wrote the book entitled ”The Theory of Option in Stocks and Shares”
In 1955, in an unpublished manuscript entitled ”Brownian Motion in the Stock Market”, a professor at the Massachusetts Institute of Technology (MIT), Paul Samuelson, 1970 Nobel Prize in Economics, showed that the asset price can be modeled by a stochastic process called the Brownian Geometric Motion
A finite element method based on Hermite polynomials to solve the nonlinear problem in the non-divergent form, in a domain with fixed boundaries, was presented
Summary
The valuation of options based on stochastic processes dates back to 1877, when Charles Castelli wrote the book entitled ”The Theory of Option in Stocks and Shares”. They are small in general, they can lead to an increase in the option price in which case the Black-Scholes pricing methodology will no longer be valid since perfect hedging is impossible.Consequentely, different models have been proposed to modify equation (1) in order to accommodate transaction costs, such as those in [11, 12, 13] In these models, the constant volatility is replaced by a modified volatility which can depend on time, on the asset price, on the option value and its derivatives. Wang [3] developed a numerical method for a nonlinear parabolic partial differential equation resulting from the pricing of European options under transaction costs Their method is based on an upwind finite difference scheme for spatial discretization and on a totally implicit scheme for time discretization.
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