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