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

2. 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.

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

4.  Coordinate transformation (Finite difference)
Strike price

5. 2. The adaptive mesh for PDE methods
Strike price

6. 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

7. 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

8. 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

9. 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