LU Decomposition with Pivoting for Option Pricing Systems
Summary
The article presents LU decomposition as a way to solve linear systems that arise when implicit finite-difference methods discretize the Black–Scholes partial differential equation. Rather than directly inverting the coefficient matrix, the method factors a pivoted matrix into lower and upper triangular matrices, then uses those factors to solve for the unknown vector. Partial pivoting reorders rows to place large column entries on the diagonal, supporting numerical stability.
It demonstrates the factorization with both SciPy’s linear algebra routines and a pure Python implementation of Doolittle’s method, and compares their outputs on the same example matrix. The article notes that production use would generally favor the library implementation for speed, while the slower hand-built version is intended to make the algorithm easier to understand. It also notes that banded systems may be solved more efficiently with specialized methods such as the Thomas algorithm. The example illustrates the computation but is not a performance benchmark or a complete option-pricing implementation.
Key ideas
- Implicit finite-difference option pricing can reduce a partial differential equation to a linear system.
- LU decomposition factors the system matrix into lower and upper triangular matrices for solving without direct inversion.
- Partial pivoting permutes rows to improve numerical stability.
- SciPy offers a concise implementation, while a pure Python version can clarify the steps at a speed cost.
- Banded matrices may be handled more efficiently with specialized solvers.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.