New optimal adaptive stepsize algorithm for solving black-scholes equation

In this paper, a new algorithm is designed based on a state feedback global error control system, Laplace trans form, order reduction method, and k-step numerical integration methods to numerically solve the Black-Scholes equation. For this purpose, the Black-Scholes equation is converted into a first-order system of ordinary differ ential equations by using the Laplace transform and order reduction method. Also, a new robust linear optimal adaptive global error control dynamic for designing an adaptive time variant step size sequence is modeled and a corresponding optimal control law based on robust and optimal eigenvalue assignment is designed. The proposed optimal control law guarantees the absolute stability of the implemented k-step numerical integrator methods.  Finally, the transformed approximate solution of the Black-Scholes equation has been obtained using the Stefhest inverse Laplace transformation algorithm. The simulation examples show that the optimal control of global error under a given tolerance level, the guarantee of absolute stability, and the best approximation of sensitivity analysis indexes for the proposed approximate solution of the Black-Scholes equation is among the important advantages of the proposed method.

کلیدواژه ها:

Black-Scholes equation ، Laplace transform ، Global error ، Numerical integration ، State feedback ، Adaptive step size ، Eigenvalues as signment ، Certainty matrix