Incremental computation of block triangular matrix exponentials with application to option pricing
For practitioners in quantitative finance using polynomial diffusion models, this work provides an incremental method to reduce the cost of repeated matrix exponential computations during discretization.
The paper addresses the problem of incrementally computing matrix exponentials of a sequence of nested block triangular matrices, arising in option pricing under polynomial diffusion models. The proposed algorithm, based on scaling and squaring with reuse of intermediate quantities, achieves computational savings compared to recomputing from scratch.
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.