High-accuracy PDE Method for Financial Derivative Pricing Shan Zhao and G. W. Wei Department of Computational Science National University of Singapore,

1. Introduction Major numerical approaches for option pricing Binomial tree model Finite difference method Monte Carlo simulation Simple, flexible, and convergent The speed of convergence usually slow.

Towards accuracy improvements:  The adaptive mesh model (Trinomial model) Strike price  Reason  Local  Adaptive

 Coordinate transformation (Finite difference) Strike price

2. The adaptive mesh for PDE methods Strike price

Numerical valuation of European call options by using FD Uniform mesh Adaptive mesh L error 1.15E-3 3.65E-4 L2 error 2.99E-4 9.40E-5 Execution time (s) 4.47E-4 9.48E-4

Handling complex boundary conditions 3. Discrete singular convolution (DSC) algorithm Handling complex boundary conditions Accuracy Approximation style Examples PDE Methods Flexible Low Finite difference Local Inflexible High Spectral Global Flexible High DSC Unified

Numerical valuation of European call options by using DSC Uniform mesh Adaptive mesh L error 7.93E-4 2.22E-8 L2 error 2.38E-4 6.73E-9 Execution time (s) 4.85E-2 1.24E-1 >4

4. Conclusion I. To achieve more accurate valuation Higher resolution meshes Higher order methods II. Higher order PDE methods for financial derivative pricing Rarely used High accuracy and efficient Promising