The Use of Numeraires in Multi-dimensional Black-Scholes Partial Differential Equations

TL;DR

This paper explores the mathematical theory behind the numeraire technique in multi-dimensional Black-Scholes PDEs, demonstrating how it simplifies complex derivative pricing by reducing risk sources and dimensionality.

Contribution

It provides a theoretical framework for numeraire methods within PDE theory and illustrates their application through five concrete pricing problems.

Findings

Numeraire technique reduces risk sources in option pricing.

It simplifies PDEs by lowering the dimension of the state space.

The approach is effective for complex derivatives with multiple risks.

Abstract

The change of numeraire gives very important computational simplification in option pricing. This technique reduces the number of sources of risks that need to be accounted for and so it is useful in pricing complicated derivatives that have several sources of risks. In this article, we considered the underlying mathematical theory of numeraire technique in the viewpoint of PED theory and illustrated it with five concrete pricing problems. In the viewpoint of PED theory, the numeraire technique is a method of reducing the dimension of status spaces where PDE is defined.