A Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes Equations

TL;DR

This paper introduces a multiplicative transformation method that preserves the form of multi-dimensional Black-Scholes equations, effectively reducing the number of risk sources in financial models by leveraging invariance properties under variable transformations.

Contribution

It demonstrates that a specific multiplicative change of variables maintains the equation's form and reduces dimensionality, with practical examples in financial pricing.

Findings

Reduces risk sources in multi-dimensional models

Preserves Black-Scholes equation form under transformation

Applicable to various financial pricing problems

Abstract

We study a method of reducing space dimension in multi-dimensional Black-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the Black-Scholes partial differential equation reserves the form of Black-Scholes partial differential equation and reduces the space dimension. We show that this transformation can reduce the number of sources of risks by two or more in some cases by giving remarks and several examples of financial pricing problems. We also present that the invariance of the form of Black-Scholes equations is based on the invariance of the form of parabolic equation under a change of variables with the linear combination of variables.