1. Finite Difference Schemes Dr. DAI Min

2. Type of finite difference scheme Explicit scheme – Advantage There is no need to solve a system of algebraic equations Easy for programming – Disadvantage: conditionally convergent Implicit scheme – Fully implicit scheme: first order accuracy – Crank-Nicolson scheme: second order accuracy

3. Explicit scheme European put option: Lattice:

4. Explicit scheme (continued)

7. Monotone scheme

8. Explicit scheme for a transformed equation Transformed Black-Scholes equation:

9. Explicit scheme for a transformed equation

10. Explicit scheme for a transformed equation (continued)

12. Equivalence of explicit scheme and BTM

13. Equivalence of explicit scheme and BTM (continued)

14. Why use implicit scheme? Explicit scheme is conditionally convergent

15. Fully implicit scheme

16. Fully implicit scheme (continued)

17. Matrix form of an explicit scheme

18. Monotonicity of the fully implicit scheme

19. Second-order scheme: Crank-Nicolson scheme

20. Crank-Nicolson scheme in matrix form

21. Convergence of Crank-Nicolson scheme The C-N scheme is not monotone unless  t/h 2 is small enough. Monotonicity is sufficient but not necessary The unconditional convergence of the C-N scheme (for linear equation) can be proved using another criterion (see Thomas (1995)). Due to lack of monotonicity, the C-N scheme is not as stable/robust as the fully implicit scheme when dealing with tough problems.

22. Iterative methods for solving a linear system

23. Linearization for nonlinear problems

24. Newton iteration

25. Handling non-smooth terminal conditions C-N scheme has a better accuracy but is unstable when the terminal condition is non-smooth. To cure the problem – Rannacher smoothing – Smoothing the terminal value condition

26. Upwind (upstream) treatment

27. An example for upwind scheme in finance

28. Artificial boundary conditions Solution domain is often unbounded, but implicit schemes should be restricted to a bounded domain – Truncated domain – Change of variables Artificial boundary conditions should be given based on – Properties of solution, and/or – PDE with upwind scheme

29. Examples European call options CIR model for zero coupon bond

30. CIR models (continued) Method 1: confined to [0,M] Method 2: a transformation

31. Test of convergence order

32. Test of convergence order (alternative method)

33. An example: given benchmark values

34. An example: no benchmark values