Finite Difference Schemes

Dr. DAI Min

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

Explicit scheme• European put option:

• Lattice:

Explicit scheme (continued)

Explicit scheme (continued)

Explicit scheme (continued)

Explicit scheme (continued)

• Monotone scheme

Explicit scheme for a transformed equation

• Transformed Black-Scholes equation:

Explicit scheme for a transformed equation

Explicit scheme for a transformed equation (continued)

Explicit scheme for a transformed equation (continued)

Equivalence of explicit scheme and BTM

Equivalence of explicit scheme and BTM (continued)

Why use implicit scheme?

• Explicit scheme is conditionally convergent

Fully implicit scheme

Fully implicit scheme (continued)

Matrix form of an explicit scheme

Monotonicity of the fully implicit scheme

Second-order scheme: Crank-Nicolson scheme

Crank-Nicolson scheme in matrix form

Convergence of Crank-Nicolson scheme

• The C-N scheme is not monotone unless t/h2 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.

Iterative methods for solving a linear system

Linearization for nonlinear problems

Newton iteration

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

Upwind (upstream) treatment

An example for upwind scheme in finance

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

Examples

• European call options

• CIR model for zero coupon bond

CIR models (continued)

• Method 1: confined to [0,M]

• Method 2: a transformation

Test of convergence order

Test of convergence order (alternative method)

An example: given benchmark values

An example: no benchmark values