NUMERICAL SOLUTION WITH HIGHER ORDER ACCURACY FOR OPTION PRICING WITH STOCHASTIC VOLATILITY USING A GEOMETRICAL TRANSFORMATION

‎In this paper using a geometrical transformation we propose a‎ ‎new compact finite difference (CFD) method based on the alternating direction implicit (ADI) approach for solving‎ ‎Heston equation that plays an important role in financial option‎ ‎pricing theory‎. ‎A feature of this time-dependent two-dimensional‎ ‎convection-diffusion-reaction equation is the presence of a mixed‎ ‎spatial-derivative term which stems from the correlation between two underlying‎ ‎stochastic processes for the asset price and its variance‎. ‎Proposed method leads to a‎ ‎system of linear equations involving banded matrices and‎ ‎the rate of convergence of the method is of order $O(k^2+h_1^8+h_2^8)$ where $k$‎, ‎$h_1$ and $h_2$ are time and space‎ ‎step-sizes‎, ‎respectively‎. ‎Stability analysis of the method is investigated by‎ ‎the matrix method‎. ‎Numerical results obtained by the‎ ‎proposed method imply that our method is effective and applicable for solving such problems‎.