# Incremental computation of block triangular matrix exponentials with application to option pricing

## TL;DR

This paper presents an efficient incremental algorithm for computing block triangular matrix exponentials, tailored for option pricing in polynomial diffusion models, by reusing computations across stages.

## Contribution

It introduces a novel incremental approach that computes matrix exponentials block by block, optimizing performance in financial modeling applications.

## Key findings

- Efficient reuse of intermediate calculations reduces computational cost.
- Algorithm effectively handles nested block triangular matrices in option pricing.
- Demonstrates practical applicability in polynomial diffusion models.

## Abstract

We study the problem of computing the matrix exponential of a block triangular matrix in a peculiar way: Block column by block column, from left to right. The need for such an evaluation scheme arises naturally in the context of option pricing in polynomial diffusion models. In this setting a discretization process produces a sequence of nested block triangular matrices, and their exponentials are to be computed at each stage, until a dynamically evaluated criterion allows to stop. Our algorithm is based on scaling and squaring. By carefully reusing certain intermediate quantities from one step to the next, we can efficiently compute such a sequence of matrix exponentials.