The Crank-Nicolson method is a finite difference method used for numerically solving the heat and related equations. It is a second-order method in time, and is numerically stable. It involves solving a tridiagonal system of linear equations. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century.

The Crank-Nicolson method involves taking the derivative half way between the beginning and the end of the time space. It is hence an average between a fully implicit and fully explicit model of PDE's. This is where the second-order convergence comes from, because essentially the first-order error term drops out from the averaging.

Because a number of other phenomena can be modeled with the heat equation (often called the diffusion equation in financial mathematics), the Crank-Nicolson method has been applied to those areas as well. Particularly, the Black-Scholes option pricing model's differential equation can be transformed into the heat equation, and thus option pricing numerical solutions can be obtained with the Crank-Nicolson method. The importance of that comes from the extensions of the option pricing model that are not able to be represented with a closed form analytic solution; they can still offer numerical solutions.

The method

or, for a uniform grid in two dimensions:

$u\_\{j,k\}^\{n+1\}\; =\; u\_\{j,k\}^n\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{a\; \backslash Delta\; t\}\{(\backslash Delta\; x)^2\}\; \backslash left+\; u\_\{j-1,k\}^\{n+1\}\; +\; u\_\{j,k+1\}^\{n+1\}\; +\; u\_\{j,k-1\}^\{n+1\}\; -\; 4u\_\{j,k\}^\{n+1\})\; +\; (u\_\{j+1,k\}^\{n\}\; +\; u\_\{j-1,k\}^\{n\}\; +\; u\_\{j,k+1\}^\{n\}\; +\; u\_\{j,k-1\}^\{n\}\; -\; 4u\_\{j,k\}^\{n\})\backslash right$

See also

• Financial mathematics
• partial differential equations

References

• Crank J. and Nicolson P. (1947) "A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type". Proceedings of the Cambridge Philosophical Society 43, 50–64.
• Wilmott P., Howison S., Dewynne J. (1995) The Mathematics of Financial Derivatives:A Student Introduction. Cambridge University Press.
• Fitzpatrick R. (2003) The Crank-Nicolson scheme. Retrieved May 4, 2005.
• Pitman E. Bruce. (1999) Parabolic equations. Retrieved May 4, 2005.
• O'Connor J. and Robertson E. (2000) John Crank. Retrieved May 4, 2005.
• O'Connor J. and Robertson E. (2000) Phyllis Nicolson. Retrieved May 4, 2005.

External links

• Module for Parabolic P.D.E.'s

Mathematical finance | Numerical linear algebra