Articlehybrid2014

Cubic Spline Method for a Generalized Black‐Scholes Equation

15 citations0/0 files storedMathematical Problems in Engineering

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Stochastic processes and financial applications Numerical methods for differential equations Differential Equations and Numerical Methods

Abstract

We develop a numerical method based on cubic polynomial spline approximations to solve a a generalized Black‐Scholes equation. We apply the implicit Euler method for the time discretization and a cubic polynomial spline method for the spatial discretization. We show that the matrix associated with the discrete operator is an M‐matrix, which ensures that the scheme is maximum‐norm stable. It is proved that the scheme is second‐order convergent with respect to the spatial variable. Numerical examples demonstrate the stability, convergence, and robustness of the scheme.

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1973The Pricing of Options and Corporate Liabilities 1979Option pricing: A simplified approach 1994Option pricing: Mathematical models and computation 1977The valuation of warrants: Implementing a new approach 2000Far Field Boundary Conditions for Black--Scholes Equations 1997Numerical Methods in Finance 1988The Use of the Control Variate Technique in Option Pricing 1982A More Accurate Finite Difference Approximation for the Valuation of Options

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