The Finite-difference Numerical Method Of European Call Option Pricing

In the complex and volatile financial market, financial derivative products, with such great functions as leverage and hedging against risks, are popular among investors and draw many mathematicians and financial economists attention to study them extensively. To effectively manage the risks effectively, the products must be correctly valued. Taking the European call-option as an object, this paper studies the numerical solutions of two kinds of Black-Scholes models that one without dividends and the other with dividends.In the third chapter, we discuss the numerical solutions of Black-Scholes mod-els without dividends. Firstly, we transform the Black-Scholes equation into a stan-dard parabolic partial differential equation equivalently. Secondly, through the Lagrange interpolation-polynomial, we construct a seven-point difference GMRES formula, with the order O(h6+Δτ3), and with the formula we construct a iterative equation. Meanwhile, we show that the difference formula is unconditionally stable and convergence by Fourier analysis. Lastly, we give a numerical example to verify the efficiency of the difference.In the fourth chapter, we study the Black-Scholes model with dividends by the finite-difference method. Firstly, we equivalently transform the Black-Scholes equation into a constant coefficient parabolic partial differential equation. Secondly, utilizing the Lagrange interpolation-polynomial, we construct a five-point difference GMRES scheme with the order O(h4+Δτ3). Again we show that the difference is unconditionally stable and convergence by Fourier analysis. Lastly, we give a numerical example to verify the efficiency of the difference as well.