A0578
Title: Numerical methods for Black-Scholes-type PDEs with time and space-varying coefficients in 1D and 2D
Authors: Dona Amanda Munasinghe - University of Wollongong (Australia) [presenting]
Mariano Rodrigo - University of Wollongong (Australia)
Abstract: The purpose is to explore numerical methods for solving 1D and 2D Black-Scholes-type partial differential equations (PDEs), which are essential in modeling derivative pricing. These equations involve parameters like volatility, interest rates, and asset prices, which, in realistic market scenarios, particularly in weather, energy, and climate derivatives, often vary over time or depend nonlinearly on the underlying assets. Three numerical schemes are evaluated for handling such variable coefficients: semi-Lagrangian scheme (SLS), Crank-Nicolson (CN), and a hybrid SLS-CN for 1D PDEs; and alternating direction implicit (ADI), SLS, and a hybrid SLS-ADI for 2D PDEs. In 1D cases, the hybrid SLS-CN method balances accuracy and computational efficiency, outperforming CN while maintaining precision. For 2D problems, the SLS-ADI hybrid proves most effective, merging ADIs dimensional splitting with SLSs adaptability to irregular grids. A general 2D PDE is used as a representative example, since multidimensional problems typically introduce cross-derivatives and nonlinear interactions that test the robustness of numerical schemes. Pure ADI methods handle separable operators well but struggle with complex cases, while standalone SLS methods experience high interpolation costs. The hybrid approach mitigates both issues, offering a scalable, accurate, and efficient solution, particularly for pricing weather and climate derivatives with time and asset-dependent dynamics.
