Numerical solutions of Black-Scholes integro-differential equations with convergence analysis

Authors: AZIM RIVAZ, MAHMOUD MOHSENI MOGHADAM, SAMANEH BANIASADI

Abstract: Stochastic integro-differential equations are obtained when we consider prices jump in financial modelling. In this paper, these equations are solved numerically by applying the two-dimensional Tau method with ordinary bases. Next, the numerical solutions of the equations above are investigated by the ordinary bases to the Hermitian one. Moreover, we provide an error analysis for the Tau method with ordinary bases. Also, we will prove that the errors of the approximate solutions decay exponentially in weighted ${L^{2}}$-norm. At the end, we will provide some numerical examples which show the efficiency and accuracy of the method.

Keywords: Tau method, stochastic integro-differential Black-Scholes equation, European option pricing problem, Hermitian polynomial

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